Quadratic Formula Derivation, Examples:

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Quadratic Formula Derivation: Again there are some quadratic polynomials that are not factorable at all using integral coefficients. In such a case we can always find the solution of a quadratic equation by the quadratic formula.

$a x^2+bx+c = 0$ by applying a formula known as the quadratic formula. This formula is applicable for every quadratic equation.

Derivation of Quadratic Formula

Quadratic Formula Derivation

The standard form of a quadratic equation is

$ax^2+bx + c = 0, \;\;\;\;\;\;\;\;\;\;\;\;;a ≠ 0$

Step 1. Divide the equation by a

$x^2+\frac bax+\frac ca=0$

Step 2. Take constant term to the R.H.S

$x^2+\frac bax=-\frac ca$

Step 3. To complete the square on the L.H.S. add ${(\frac b{2a})}^2$ to both sides

$x^2+\frac bax+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac ca$

$(x+\frac b{2a})^2=\frac{b^2-4ac}{4a^2}$

$(x+\frac b{2a})=\pm\frac{\sqrt{b^2-4ac}}{2a}$

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

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

Hence the solution of the quadratic equation $ax^2+bx + c = 0$ is given by

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

which is called Quadratic formula.

Example 1: Solve the equation $6x^2 +x - 15 = 0$ by using the quadratic formula.

Solution:

Comparing the given equation with

$ax^2+bx + c = 0$ ,

we get,
$,a = 6, b = 1, c = - 15$ ,

∴ The solution is given by

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

$x=\frac{-1\pm\sqrt{1^2-4(6)(-15)}}{2(6)}$ ,

$,x=\frac{-1\pm\sqrt{361}}{12}$ ,

$x=\frac{-1\pm19}{12}$

$,i.e,\;\;\;\;\;\;\;\;\; x=\frac{-1+19}{12} \;\;\;\;\;\;\;\;\;or \;\;\;\;\;\;\;\;\; x=\frac{-1-19}{12}$ ,

Example 2: Solve the equation $8x^2 -14x - 15 = 0$ by using the quadratic formula.

Solution:

Comparing the given equation with

$ax^2+bx + c = 0$ ,

we get,
$,a = 8, b = -14, c = - 15$ ,

∴ The solution is given by

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

$x=\frac{-(-14)\pm\sqrt{(-14)^2-4(8)(-15)}}{2(8)}$ ,

$,x=\frac{14\pm\sqrt{676}}{16}$ ,

$x=\frac{14\pm26}{16}$ ,

$,i.e,\;\;\;\;\;\;\;\;\; x=\frac{14+26}{16} \;\;\;\;\;\;\;\;\;or \;\;\;\;\;\;\;\;\; x=\frac{14+26}{16}$ ,

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