15. If M and N are real roots of 4x²- 4x = 3, then compute the numerical value of M² + N²?

Answer 1

Using the quadratic formula, we can find the roots of the quadratic equation 4x² - 4x - 3 = 0.

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the roots can be found using the formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 4, b = -4, and c = -3.

Using the quadratic formula, we can find the two roots:

x = (-(-4) ± √((-4)² - 4(4)(-3))) / (2(4))
x = (4 ± √(16 + 48)) / 8
x = (4 ± √64) / 8
x = (4 ± 8) / 8

So the two possible roots are:
M = (4 + 8) / 8 = 12 / 8 = 3/2
N = (4 - 8) / 8 = -4 / 8 = -1/2

Now we can compute the value of M² + N²:

M² + N² = (3/2)² + (-1/2)²
= 9/4 + 1/4
= 10/4
= 5/2

Therefore, the numerical value of M² + N² is 5/2.