Double Roots

Question

If 16x^2 - 24x + k = 0 has two roots (a double root) equal to 3/4, 4 = ?

Answer 1

The discriminant of Ax²+Bx+C=0 is B²-4AC.  The quadratic will have a double root only when B²-4AC=0.

 

For the given equation, B²-4AC = (-24)² -  4(16)(k) = 576 - 64k.

 

Setting the discriminant equal to zero, we have 64k = 576.

 

So, k = 9.

 

Answer 2

If we use the quadratic formula, we get the two roots.  Use it to set the formula with the roots.

 

3/4 = (24 ± √(576 - 64k)) / 32 = 4

 

 

Now we use this equation to solve for k.

 

Multiply all sides of the equation by 32.

 

24 = 24 ± √(576 - 64k) = 128

 

 

Subtract 24 on all sides of the equation.

 

0 = ±√(576 - 64k) = 104

 

 

Square all sides of the equation.

 

0 = 576 - 64k = 10816

 

 

Now separate the equations into two parts.

 

576 - 64k = 0           and           576 - 64k = 10816

 

 

Solve for k from both equations.

Answer 3

16x² - 24x + k = 0

 

If we have a double root at x=3/4 then

(x-3/4)² is a factor:

 

(x- 3/4)² = x² - (3/2)x + 9/16

 

If we multiply this through by 16 we get:

16x² -24x + 9

 

k = 9