Question

write the quadratic formula for finding roots of the quadratic equation and also the nature of the roots​

Answer 1

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Answer 2

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The standard form of a quadratic equation is –

→ ax²  + bx + c = 0

Here,

1. a = Coefficient of x².
2. b = Coefficient of x.
3. c = Constant term.

Formula to find out roots:

$\tt\mapsto x_{1,2} = \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Here,

→ Discriminant = b² - 4ac

The discriminant tells us about the nature of roots.

• If Discriminant is greater than 0, roots are real and distinct.
• If the Discriminant is equal to 0, roots are equal and real.
• If the Discriminant is less than 0, no real roots exists.

Derivation of Quadratic Formula,

$\tt\implies ax^{2}+bx+c=0$

Multiplying both sides by 4a, we get,

$\tt\implies 4a^{2}x^{2}+4abx+4ac=0$

Adding b² to both sides, we get,

$\tt\implies 4a^{2}x^{2}+4abx+b^{2}+4ac=b^{2}$

$\tt\implies (2ax)^{2}+2\times (2ax)\times (b)+b^{2}=b^{2}-4ac$

$\tt\implies (2ax + b)^{2}=b^{2}-4ac$

$\tt\implies 2ax + b=\pm\sqrt{b^{2}-4ac}$

$\tt\implies 2ax =-b\pm\sqrt{b^{2}-4ac}$

$\tt\implies x_{1,2} =\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Hence derived.