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Algebra : Nature of Roots : Quadratic Equations

Checking if the roots of quadratic equation or real or imaginary

Question

The equation 2x² + 2(p + 1)x + p = 0, where p is real, always has roots that are

(1) Equal
(2) Equal in magnitude but opposite in sign
(3) Irrational
(4) Real
(5) Complex Conjugates

Correct Choice - (4). Correct Answer is The Roots are Real

Explanatory Answer

The value of the discriminant of a quadratic equation will determine the nature of the roots of a quadratic equation.

The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.

• If the value of the discriminant is positive, i.e. greater than '0', then the roots of the quadratic equation will be real.
• If the value of the discriminant is '0', then the roots of the quadratic equation will be real and equal.
• If the value of the discriminant is negative, i.e. lesser than '0', then the roots of the quadratic equation will be imaginary. The two roots will be complex conjugates of the form p + iq and p - iq.

Using this basic information, we can solve this problem as shown below.

In this question, a = 2, b = 2(p + 1) and c = p

Therefore, the disciminant will be (2(p + 1))² - 4*2*p = 4(p + 1)² - 8p
= 4[(p + 1)² - 2p]
= 4[(p² + 2p + 1) - 2p]
= 4(p² + 1)

For any real value of p, 4(p² + 1) will always be positive as p² cannot be negative for real p.

Hence, the discriminant b² - 4ac will always be positive.

When the discriminant is greater than '0' or is positive, the roots of a quadratic equation will be real.

Therefore, the answer choice is 4.

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