Question

Which of the following are solutions to the equation below?

4x² - 12x + 9 = 5
a) x = -5, x = 3/2
b) x = √5, x = 3/2
c) x = -5 + √5, x = -5 - √5
d) x = 3/2 + √5, x = 3/2 - √5

Answer 1

The quadratic equation 4x² - 12x + 9 = 5 can be solved using the quadratic formula, resulting in two solutions, x = 3/2 + √5 and x = 3/2 - √5, which corresponds to option d).

Step-by-step explanation:

The equation provided in the question is a quadratic equation of the form ax² + bx + c = 0. To solve for x, we can either factor the equation, complete the square, or use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a). Initially, let's look at the equation 4x² - 12x + 9 = 5. To start, we bring the constant on the right over to the left side to have zero on one side of the equation.

4x² - 12x + 9 - 5 = 0 ⇒ 4x² - 12x + 4 = 0 ⇒ x² - 3x + 1 = 0 (Divide by 4).

At this point, we can apply the quadratic formula. Using the general quadratic formula, the values of a, b, and c are 1, -3, and 1, respectively. After plugging in the values, we get:

x = (-(-3) ± √((-3)² - 4(1)(1))) / (2(1)) ⇒ x = (3 ± √(9 - 4)) / 2 ⇒ x = (3 ± √5) / 2

We get two potential solutions: x = 3/2 + √5 and x = 3/2 - √5. Therefore, option d) x = 3/2 + √5, x = 3/2 - √5 is correct.