Find the value of p for which the quadratic equation x2+ p(4x + p – 1) + 2 = 0 has equal roots :a)- 1,2/3b)3, 5c)1,-(4/3)d)3/4, 2Correct answer is option 'A'. Can you explain this answer?

Question

Find the value of p for which the quadratic equation x2+ p(4x + p &ndash; 1) + 2 = 0 has equal roots :a)- 1,2/3b)3, 5c)1,-(4/3)d)3/4, 2Correct answer is option 'A'. Can you explain this answer?

Answer

Answer

A quadratic equation has equal roots if it can be expressed as a 'perfect square'.
f(x) = x² + p(4x + p - 1) + 2
f(x) = x² + 4px + p² - p + 2
f(x) = x² + 4px + 4p² - (3p² + p - 2)
f(x) = (x + 2p)² - (3p² + p - 2)
For the above expression to be a perfect square,
3p² + p - 2 = 0
3p² + 3p - 2p - 2 = 0
3p(p + 1) - 2(p + 1) = 0
(p + 1)(3p - 2) = 0
=> p = -1 and ⅔

Answer

We can start by using the discriminant of the quadratic equation to find the values of p for which the equation has real roots. The discriminant is given by:

b^2 - 4ac

where a, b, and c are the coefficients of the quadratic equation. In this case, we have:

a = 1, b = p(4x), c = -p^2

Substituting these values into the discriminant formula, we get:

(p(4x))^2 - 4(1)(-p^2)

Simplifying this expression, we get:

16p^2x^2 + 4p^2

Now, for the quadratic equation to have real roots, this expression must be greater than or equal to zero. Therefore, we have:

16p^2x^2 + 4p^2 >= 0

Dividing both sides by 4p^2, we get:

4x^2 + 1 >= 0

This expression is always true, since 4x^2 is non-negative and adding 1 to it can only make it greater. Therefore, the quadratic equation has real roots for all values of p.

To find the value of p for which the quadratic equation has a repeated root, we can use the fact that a quadratic equation has a repeated root if and only if its discriminant is equal to zero. Therefore, we have:

(p(4x))^2 - 4(1)(-p^2) = 0

Simplifying this expression, we get:

16p^2x^2 + 4p^2 = 0

Dividing both sides by 4p^2, we get:

4x^2 + 1 = 0

This equation has no real solutions, since 4x^2 is always non-negative and adding 1 to it can only make it greater. Therefore, there is no value of p for which the quadratic equation has a repeated root.

In summary, the quadratic equation x^2 + p(4x - p) = 0 has real roots for all values of p, and does not have a repeated root for any value of p.

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