Prove that the roots of ax2 - bx - c - 0 are -b -b2 - 4ac2a

ax^2+bx+c=0 Dividing by a x^2+ba x+ca=0 (x+b2a)^2+ca=b^24a^2 (x+b2a)^2=b^2 4ac4a^2 x+b2a=±√(b^2 4ac)2a x= b±√(b^2 4ac)2a ...

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Mathematics

Quadratic Formula

Prove that th...

Question

Prove that the roots of $a x^{2} + b x + c = 0$ are $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

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Solution

$a x^{2} + b x + c = 0$
Dividing by $a$
$x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$
$(x + \frac{b}{2 a})^{2} + \frac{c}{a} = \frac{b^{2}}{4 a^{2}}$
$(x + \frac{b}{2 a})^{2} = \frac{b^{2} - 4 a c}{4 a^{2}}$
$x + \frac{b}{2 a} = \frac{\pm \sqrt{b^{2} - 4 a c}}{2 a}$
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

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Q. If the roots of

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

are

α, β

and the roots of

A X^{2} + B x + C = 0

are

(α - k), (β - k)

. Then

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is equal to

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Prove that the roots of ax2+bx+c=0 are −b±√b2−4ac2a

ax2+bx+c=0Dividing by ax2+bax+ca=0(x+b2a)2+ca=b24a2(x+b2a)2=b2−4ac4a2x+b2a=±√b2−4ac2ax=−b±√b2−4ac2a

Prove that the roots of $a x^{2} + b x + c = 0$ are $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$