Prove the root of equation ax2-bx-c-0 is - x- -b-b2-4ac2a

Consider the quadratic equation ax^2+bx+c ⇒ a(x^2+bax+ca)=0 ⇒ x^2+bax+ca=0 ⇒ x^2+bax= ca This can be written as #160; ⇒ x^2+2×b2ax= ca ...

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Byju's Answer

Standard X

Mathematics

Nature of Roots

Prove the roo...

Question

Prove the root of equation ${a x}^{2} + b x + c = 0$ is $\Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

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Solution

Consider the quadratic equation $a x^{2} + b x + c$
$\Rightarrow a (x^{2} + \frac{b}{a} x + \frac{c}{a}) = 0$

$\Rightarrow x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$

$\Rightarrow x^{2} + \frac{b}{a} x = - \frac{c}{a}$

This can be written as
$\Rightarrow x^{2} + 2 \times \frac{b}{2 a} x = - \frac{c}{a}$

Add $(\frac{b}{2 a})^{2}$ on both sides

$\Rightarrow x^{2} + 2 \times (\frac{b}{2 a}) x + (\frac{b}{2 a})^{2} = - \frac{c}{a} + (\frac{b}{2 a})^{2}$

This can be written as

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

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

$\Rightarrow x + \frac{b}{2 a} = \pm \sqrt{\frac{- 4 a c + b^{2}}{4 a^{2}}}$

$\Rightarrow x + \frac{b}{2 a} = \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

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

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

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Similar questions

Q. Prove that root of equation

a x^{2} + b x + c = 0

\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Q. Prove that the roots of

a x^{2} + b x + c = 0

are

\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

If b² – 4ac = 0 then The roots of the Quadratic equation ax² + bx + c = 0 are given by :

The roots of a quadratic equation $a x^{2} + b x + c = 0$ are given by $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , provided $b^{2} - 4 a c \geq 0$ .

Q. What is the nature of the roots of the quadratic equation

a x^{2} + b x + c = 0

if
(a)

b^{2} - 4 a c = 0

(b)

b^{2} - 4 a c < 0

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Prove the root of equation ax2+bx+c=0 is ⇒x=−b±√b2−4ac2a

Prove the root of equation ${a x}^{2} + b x + c = 0$ is $\Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$