Question

factor. then list the x-intercepts.

1. $9x^2 - 12x + 4$
2. $4x^2 - 20x + 25$

Explanation:

Problem 1: \(9x^2 - 12x + 4\)

Step 1: Identify the form

The quadratic \(9x^2 - 12x + 4\) is a perfect square trinomial. The general form of a perfect square trinomial is \(a^2 - 2ab + b^2=(a - b)^2\). Here, \(a^2 = 9x^2\), so \(a = 3x\), and \(b^2 = 4\), so \(b = 2\). Also, \(2ab=2\times3x\times2 = 12x\), which matches the middle term (with a negative sign).
So, \(9x^2 - 12x + 4=(3x - 2)^2\)

Step 2: Find x - intercepts

To find the x - intercepts, we set \(y = 9x^2 - 12x + 4 = 0\). Since \(9x^2 - 12x + 4=(3x - 2)^2\), we set \((3x - 2)^2=0\)
Solving \(3x - 2 = 0\), we get \(3x=2\), so \(x=\frac{2}{3}\)

Step 1: Identify the form

The quadratic \(4x^2 - 20x + 25\) is a perfect square trinomial. The general form of a perfect square trinomial is \(a^2 - 2ab + b^2=(a - b)^2\). Here, \(a^2 = 4x^2\), so \(a = 2x\), and \(b^2 = 25\), so \(b = 5\). Also, \(2ab = 2\times2x\times5=20x\), which matches the middle term (with a negative sign).
So, \(4x^2 - 20x + 25=(2x - 5)^2\)

Step 2: Find x - intercepts

To find the x - intercepts, we set \(y = 4x^2 - 20x + 25 = 0\). Since \(4x^2 - 20x + 25=(2x - 5)^2\), we set \((2x - 5)^2=0\)
Solving \(2x - 5 = 0\), we get \(2x = 5\), so \(x=\frac{5}{2}\)

Answer:

Factor: \((3x - 2)^2\); x - intercept: \(x = \frac{2}{3}\)

Problem 2: \(4x^2 - 20x + 25\)