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Q.

If α, β are the roots of the equation $a x^{2} + bx + c = 0$ , then find $\propto^{2} + β^{2}$ .

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a

\frac{b^{2} + 4 ac}{4 a}

b

\frac{{4 ac - b}^{2}}{2 a}

c

\frac{b^{2} - 2 ac}{a^{2}}

d

0

answer is C.

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Detailed Solution

The given equation is a quadratic equation.
We will use the relation of the zeroes and coefficients of the variables of the quadratic equation.
The sum of zeroes of a quadratic equation is equal to the negative ratio of the coefficient of x, and the coefficient of

x^{2}

.
We can observe that in

{ax}^{2} + bx + c = 0

, the coefficient of x is b, and the coefficient of

x^{2}

is a.
Therefore, we get α+β=−

\frac{b}{a}

The product of zeroes of a quadratic equation is equal to the ratio of the constant, and the coefficient of

x^{2}

.
We can observe that in

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

, the constant is c, and the coefficient of

x^{2}

is a.
Therefore, we get αβ=

\frac{c}{a}

Now, we will use algebraic identity to get the required value.
We know that the square of the sum of two numbers a and b is given by the algebraic identity

{(a + b)}^{2} = a^{2} + b^{2} + 2 ab

.
Substituting the zeroes a=α and b=β in the algebraic identity, we get
⇒

{(\propto + β)}^{2} = α^{2} + β^{2} + 2 αβ

Substituting α+β=−

\frac{b}{a}

and αβ=

\frac{c}{a}

in the equation, we get
⇒

{(- \frac{b}{a})}^{2} = α^{2} + β^{2} + 2 \frac{c}{a}

Multiplying the terms, we get
⇒

\frac{b^{2}}{a^{2}}

=

α^{2} + β^{2} + 2 \frac{c}{a}

Subtracting

2 \frac{c}{a}

from both sides, we get
⇒

\frac{b^{2}}{a^{2}} - 2 \frac{c}{a}

=

α^{2} + β^{2} + 2 \frac{c}{a} - 2 \frac{c}{a}

⇒

\frac{b^{2}}{a^{2}} - 2 \frac{c}{a}

=

α^{2} + β^{2}

Taking the L.C.M., we get
⇒

\frac{b^{2} - 2 ac}{a^{2}}

=

α^{2} + β^{2}

∴We get the value of

α^{2} + β^{2}

as

\frac{b^{2} - 2 ac}{a^{2}}

.
Thus, the correct option is option (3).

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