Question

If the zeroes of a quadratic polynomial p(x) = ax? + x + a are equal, then the value of a is​

Answer 1

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°°° Solution °°°

Correct option is C

c and a have the same sign

Given that the zeros of the quadratic polynomial a x 2 +bx+c,c

= 0 are equal.

=> Value of the discriminant(D) has to be zero.

=> b 2 − 4ac=0

=> b 2

= 4ac

Since. L.H.S b 2

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cannot be negative, thus, R.H.S. can also be never negative.

Therefore, a and c must be of the same sign.

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Answer 2

Given:

A quadratic polynomial p(x) = $ax{2} + x + a.$ having equal roots.

To Find:

The value of a if the roots of the polynomial are equal.

Solution:

The given question can be solved by using the properties of quadratic equations.

1. Consider a quadratic equation$ax^{2} + bx +c =0$.

• The discriminant of a quadratic equation is given by the formula$\sqrt{b^{2}-4ac }$. where a, b, and c are the coefficients in the quadratic polynomial. The Discriminant is also denoted by D.
• For a quadratic equation to have real and distinct roots, the condition is D > 0.
• For a quadratic equation to have equal and real roots, the condition is D = 0.

• For a quadratic equation to have imaginary/unreal roots, the condition is D < 0.

2. In the given question it is mentioned that the roots are equal, Therefore, the condition used is D = 0.

=> $ax^{2}+x+a$ is the given polynomial. On comparing with the polynomial form ($ax^{2}+bx+c$) the values of a b and c are a, 1, a respectively. ( a and c are equal in this case).

3. Substitute the values of a, b, c in the discriminant form,

=> D = $\sqrt{ b^{2} - 4ac$ = 0,

=> 1*1 - 4 * a * a = 0,

=>$4a^{2}= 1$,

=>$a = \sqrt{1/4}$,

=> a = +1/2 or a = -1/2.

Therefore, the possible values of a are +(1/2) and -(1/2).