Quadratic equation

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Dalam aljabar, persamaan kuadrat (dari bahasa Latin quadratus untuk "persegi") adalah persamaan yang memiliki bentuk

$            {\displaystyle         a        x^{            2          }                b        x                c        =        0      }        \{\backslash \; displaystyle\; axe\; ^\; \{2\}\; bx\; c\; =\; 0\}\;  ÂÂ$

where x represents the unknown, and a , and c represent numbers that are known so that a . If a = 0 , then the equation is linear, rather than squared. The number a , b , and c is coefficient equations, and can be distinguished by calling them, respectively, squared coefficients , linear and coefficients free term .

Since the quadratic equation involves only the unknown, it is called "univariate". The quadratic equation contains only the power of x which is a non-negative integer, and is therefore a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.

Quadratic equations can be solved by a process known in American English as a factorization and in various other types of English as factoring , by finishing squared, using quadratic formulas, or by graphs. The solution to a problem equivalent to the quadratic equation was known in early 2000 BC.

Video Quadratic equation

Memecahkan persamaan kuadrat

Quadratic equations with real or complex coefficients have two solutions, called roots . Both of these solutions may or may not be different, and they may or may not be real.

Factoring with review

It is possible to express the squared equation of ax ² bx c = 0 > ( px q ) ( rx s ) = 0 . In some cases, it is possible, with a simple check, to determine the values ​​ p , q , r, and s px = 0 or i> rx s = 0 . Solving these two linear equations gives the square root.

For most students, factoring with inspection is the first method to solve the quadratic equations with which they are exposed. If one is given a quadratic equation in the form x ² bx c = 0 , the factorization sought after ( x q ) ( x s ) , and we must find two numbers q and s that add up to b and whose products are c (these are sometimes called "Vieta rules" and are associated with Vieta formulas). For example, x ² 5 x 6 factors as ( x 3) ( x 2) . The more common case where a is not the same 1 can require great effort in trial and error and guess-and-check, assuming that it can be taken into account at all with the inspection.

Except for special cases such as where b = 0 or c = 0 , factoring with inspection only works to quadratic equations that have rational roots. This means that most quadratic equations appearing in practical applications can not be solved by factoring by inspection.

Complete box

Proses menyelesaikan alun-alun memanfaatkan identitas aljabar

${\ displaystyle x ^ {2} 2hx h ^ {2} = (x h) ^ {2},}  ÂÂ$

which represent well-defined algorithms that can be used to solve any quadratic equations. Starting with the quadratic equation in standard form, ax ² bx c = 0

1. Divide each side by a , the coefficient of the quadratic term.
2. Subtract the constant terms c / a from both sides.
3. Add half squares of b / a , coefficient x to both sides. It "completes the square", turning the left side into a perfect square.
4. Write the left side as square and simplify the right side if necessary.
5. Produces two linear equations by equating the square root of the left side with the positive and negative square root of the right side.
6. Complete two linear equations.

Kami mengilustrasikan penggunaan algoritme ini dengan menyelesaikan 2 x ² 4 x - 4 = 0

${\ displaystyle 1) \ x ^ {2} 2x-2 = 0}  ÂÂ$

${\ displaystyle 2) \ x ^ {2} 2x = 2}  ÂÂ$

${\ displaystyle 3) \ x ^ {2} 2x 1 = 2 1}  ÂÂ$

${\ displaystyle 4) \ \ kiri (x 1 \ right) ^ {2} = 3}  ÂÂ$

${\ displaystyle 5) \ x 1 = \ pm {\ sqrt {3}}}  ÂÂ$

${\ displaystyle 6) \ x = -1 \ pm {\ sqrt {3}}}  ÂÂ$

The plus-minus symbol "Â ±" indicates that both x = -1 ? 3 and x = -1 - ? 3 is the solution of the quadratic equation.

Quadratic formulas and their derivatives

Completing the box can be used to lower the general formula to solve the squared equation, called squared formula. Mathematical evidence will now be summarized briefly. It can be easily seen, with a polynomial expansion, that the following equation is equivalent to the squared equation:

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{b}{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂaÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âb^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â-Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â4Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂaÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂcÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â4Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âa^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â.Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\{\backslash \; Displaystyle\; \backslash \; left\; (x\; \{\backslash \; frac\; \{b\}\; \{2a\}\}\; \backslash \; right)\; ^\; \{2\}\; =\; \{\backslash \; frac\; \{b\; ^\; \{2\}\; -4ac\}\; \{4a\; ^\; \{2\}\}\}.\}\; Â\; Â$

Mengambil akar kuadrat dari kedua sisi, dan mengisolasi x , memberikan:

${\ displaystyle x = {\ frac {-b \ pm {\ sqrt {b ^ {2} -4ac \}}} {2a}}.}  ÂÂ$

Some sources, especially older ones, use alternative parametric quadratic equations such as ax ² 2 bx c = 0 or ax ² - 2 < i> = 0 , where b has a magnitude of one-half of the more common, possibly with the opposite sign. This results in a slightly different form for the solution, but rather the equivalent.

A number of alternative derivations can be found in the literature. These proofs are simpler than standard resolving quadratic methods, are interesting applications of other techniques often used in algebra, or offer insight into other areas of mathematics.

Sebuah rumus kuadrat yang kurang dikenal, seperti yang digunakan dalam metode Muller, dan yang dapat ditemukan dari formula Vieta, memberikan akar yang sama melalui persamaan:

${\ displaystyle x = {\ frac {-2c} {b \ pm {\ sqrt {b ^ {2} -4ac}}}}.}  ÂÂ$

One property of this form is that it generates a valid root when a = 0 , while the other root contains the division by zero, because when a = 0 , the quadratic equation becomes a linear equation, which has one root. In contrast, in this case, the more general formula has a division by zero for one root and an indefinite form 0/0 for other roots. On the other hand, when c = 0 , the more general formula produces two correct roots while this form produces zero root and the indefinite 0/0 .

Reduce the quadratic equation

Sometimes it's easy to reduce quadratic equations so the main coefficient is one. This is done by dividing both sides by a , which is always possible because a is not zero. This results in the reduction of squared equation :

${\ displaystyle x ^ {2} px q = 0,}$

where p = b / a and q = c a This monetary equation has the same solution as the original.

Rumus kuadrat untuk solusi dari persamaan kuadrat yang berkurang, ditulis dalam bentuk koefisiennya, adalah:

${\ displaystyle x = {\ frac {1} {2}} \ kiri (-p \ pm {\ sqrt {p ^ {2} -4q}} \ kanan) ,}  ÂÂ$

atau dengan kata lain:

${\ displaystyle x = - {\ frac {p} {2}} \ pm {\ sqrt {\ left ({\ frac {p} {2}} \ kanan) ^ {2} -q}}.}  ÂÂ$

Diskriminan

Dalam rumus kuadrat, ekspresi di bawah tanda akar kuadrat disebut diskriminan dari persamaan kuadrat, dan sering diwakili menggunakan huruf besar D atau delta Yunani huruf besar:

${\ displaystyle \ Delta = b ^ {2} -4ac.}  ÂÂ$

So the roots are different if and only if the discriminant is not zero, and the root is real if and only if discriminant is not negative.

Geometric interpretation

Function f ( x ) = ax ² bx c is a quadratic function. The graph of each quadratic function has the same general form, called a parabola. The location and size of the dish, and how it opens, depends on the value of a , b , and < i> c . As shown in Figure 1, if a & gt; 0 , the parabola has a minimum point and is open up. If a & lt; 0 , the parabola has the maximum point and is open down. The extreme point of the parabola, whether minimum or maximum, corresponds to the vertex. x -coordinate of the point will be located in $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...}$
               -                  b        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,
               2                  a        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,       ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                                {\ displaystyle \ scriptstyle x = {\ tfrac {-b} {2a}}} , and y -coordinate from that point can be found by replacing x -value into the function. y -intercept is located at the point (0, c ) .

The solution of the quadratic equation ax ² bx c = 0 according to the function root f ( x ) = ax ² bx c , since they are x values ​​for f ( x ) = 0 . As shown in Figure 2, if a , b , and c is a real number and the domain f is the set of real numbers, then the root f is exactly x -the coordinates of the points where the graph touches x -axis. As shown in Figure 3, if the discriminant is positive, the graph touches x -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch x -axis.

Quadratic factorization

Syarat

${\ displaystyle x-r}  ÂÂ$

merupakan faktor dari polinomial

$            {\displaystyle         a        x^{            2          }                b        x                c      }        \{\backslash \; displaystyle\; axe\; ^\; \{2\}\; bx\; c\}\;  ÂÂ$

jika dan hanya jika r adalah akar dari persamaan kuadrat

$            {\displaystyle         a        x^{            2          }                b        x                c        =        0.      }        \{\backslash \; displaystyle\; axe\; ^\; \{2\}\; bx\; c\; =\; 0.\}\;  ÂÂ$

Ini mengikuti dari rumus kuadrat itu

$            {\displaystyle         a        x^{            2          }                b        x                c        =        a                  \left(                      x            -                          \frac{                  -                  b                                                        \sqrt{                      b^{                          2                        }                      -                      4                      a                      c                    }                                  }{                  2                  a                }                                \right)                          \left(                      x            -                          \frac{                  -                  b                  -                                      \sqrt{                      b^{                          2                        }                      -                      4                      a                      c                    }                                  }{                  2                  a                }                                \right)                .      }        \{\backslash \; displaystyle\; ax\; ^\; \{2\}\; bx\; c\; =\; a\; \backslash \; left\; (x\; -\; \{\backslash \; frac\; \{-b\; \{\backslash \; sqrt\; \{b\; ^\; \{2\}\; -4ac\}\}\; \}\; \{2a\}\}\; \backslash \; right)\; \backslash \; left\; (x\; -\; \{\backslash \; frac\; \{-b\; -\; \{\backslash \; sqrt\; \{b\; ^\; \{2\}\; -4ac\}\}\}\; \{2a\}\}\; \backslash \; right).\}\;  ÂÂ$

Dalam kasus khusus b ² = 4 ac di mana kuadrat hanya memiliki satu akar yang berbeda ( yaitu diskriminan adalah nol), polinomial kuadrat dapat diperhitungkan sebagai

${\ displaystyle ax ^ {2} bx c = a \ left (x {\ frac {b} {2a}} \ right) ^ {2}.}  ÂÂ$

Graphing untuk akar sebenarnya

For much of the 20th century, graphs are rarely mentioned as methods for solving quadratic equations in high school or college algebra texts. Students learn to solve quadratic equations by factoring, solving squares, and applying quadratic formulas. Recently, graphic calculators have become common in schools and graphical methods have begun to appear in textbooks, but generally not overly emphasized.

Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a chart y = f ( x ) , the ability to scaled the graph precisely to the surface dimensions of the graph, and the recognition that when f ( x ) = 0 , x is the solution to the equation. The skills required to solve quadratic equations on a calculator actually apply to find the true root of any arbitrary function.

Because the arbitrary function can cross x -axis at many points, the graphing calculator generally requires one to identify the desired root by positioning the cursor at the "guess" value for the root. (Some graphing calculators require root bracketing on both sides of the zero.) The calculator then proceeds, with iterative algorithms, to fix the approximate root position to the limits of calculator accuracy.

Avoid loss of significance

Although the quadratic formula provides the right solution, the result is not correct if the real number is estimated during the calculation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "real" in many programming languages). In this context, the quadratic formula is not fully stable.

This occurs when the root has a different order of magnitudes, or, equivalently, when b ² and b ² - 4 ac > very close. In this case, a similar two-figure reduction would result in a loss of significance or catastrophic cancellation at smaller roots. To avoid this, the smaller root in magnitude, r , can be counted as ${\ displaystyle (c/a)/R}$ where R is a larger root in magnitude.

The second form of cancellation may occur between the terms b ² and 4 ac of the discriminant, ie when two roots are very close. This can result in the loss of up to half of the important numbers correct at the root.

Maps Quadratic equation

Examples and apps

Rasio emas ditemukan sebagai solusi positif dari persamaan kuadrat ${\ displaystyle x ^ {2} -x-1 = 0.}  ÂÂ$

The equations of circles and other conic sections - ellipses, parabolas, and hyperboles - are quadratic equations in two variables.

Given a cosine or sinus from an angle, finding a cosine or sinus from a half-angle involves solving a quadratic equation.

The process of simplifying expressions involving the square root of expressions involving the square roots of other expressions involves finding two solutions to quadratic equations.

Descartes's theorem states that for every four circles of kiss (which are mutually intersect), their fingers satisfy a certain quadratic equation.

The equation given by Fuss' theorem, gives the relationship between the radius of a bicentric rectangular circle, the circumscribed circle of radius, and the distance between the center of the circle, can be expressed as a quadratic equation of the distance between centers of two circles in which case their fingers are wrong one solution. Another solution of the same equation in the case of the relevant radius provides the distance between the center of the finite circle and the excircle center of the ex-tangential rectangle.

src: saylordotorg.github.io

History

The mathematician of Babylonia, in early 2000 BC (displayed on the Old Babylon clay tablet) can solve problems related to the area and the rectangular side. There is evidence showing this algorithm since the Third Dynasty of Ur. In modern notation, the problem usually involves the completion of a pair of simultaneous equations of the form:

${\ displaystyle x y = p, \ \ xy = q,}$

yang setara dengan pernyataan bahwa x dan y adalah akar dari persamaan:

${\ displaystyle z ^ {2} q = pz.}  ÂÂ$

The steps given by the Babylonian scholars to solve the rectangular problem above, in terms of x and y , are as follows:

1. Calculate half of p .
2. Create a result box.
3. Reduce q .
4. Find square root (positive) using square squared.
5. Add together step (1) and (4) step results in providing x . In modern notation, this means computing ${\ displaystyle x = {\ frac {p} {2}} {\ sqrt {\ left ({\ frac {p} {2}} \ right ) ^ {2} -q}}.} Â Â$

Geometric methods are used to solve quadratic equations in Babylon, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains solutions to two-term quadratic equations. Babylonian mathematicians from about 400 BC and Chinese mathematicians from about 200 BC used geometric dissection methods to solve quadratic equations with positive roots. The rules for quadratic equations are given in The Nine Chapters on the Mathematical Art , a Chinese treatise on mathematics. This initial geometric method does not seem to have a general formula. Euclid, the Greek mathematician, produced a more abstract method of geometry around 300 BC. With a pure geometric approach, Pythagoras and Euclid created a general procedure for finding solutions to quadratic equations. In his Arithmetica , Greek mathematician Diophantus solved a quadratic equation, but gave only one root, even when both roots were positive.

In 628 AD, Brahmagupta, an Indian mathematician, provided the first explicit solution (although still not entirely common) of the squared equation ax ² bx = c as follows: "For absolute numbers multiplied by four squares [coefficients], add squares of the medium term [coefficient]; same squared roots, , divided twice [squared coefficient] is its value. "(Brahmasphutasiddhanta , Colebrook translation, 1817, page 346) This is equivalent to:

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\sqrt{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â4Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂaÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂcÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âb^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â-Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂbÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂaÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}}$

Source of the article : Wikipedia