Question

Find the values of p for which the quadratic equation 4x2 + px + 3 = 0 has equal roots.​

Answer 1

p = ±4√3

Explanation:

The discriminant of quadratic equation ax²+bx+c = 0 is ...

d = b² -4ac

In your quadratic, its value is ...

d = p² -4(4)(3) = p² -48

The discriminant will be zero when the quadratic has equal roots. In that case, the values of p are found by ...

d = 0

p² -48 = 0

p² = 48 . . . . . add 48

p = ±√48 . . . . take the square root

p = ±4√3 . . . . simplify

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Another way to get there is by looking at the factoring of a quadratic with equal roots:

(ax +b)² = 0 = a²x² +2ab +b²

Comparing to the given quadratic, we find ...

a² = 4 ⇒ a = ±2 . . . . . . coefficient of x²

b² = 3 ⇒ b = ±√3 . . . . constant

p = 2ab = 2(±2)(±√3) . . . . coefficient of x

p = ±4√3

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The attachment shows the two different values of p give equations with one solution (each).

Answer 2

Explanation:

You need to concern yourself with the +3. That will determine what the equal roots are.

(2x + √3)^2

Now expand this to see what p is. (Kind of a nasty question if you ask me).

4x^2 + 2x*√3 + (√3)^2

4x^2 + 2√3 x + 3 also possible is

4x^2 - 2√3 x + 3

So the answer is p = 2√3

The answer could also be p = -2√3