Question

1. $x^2 - 4x + 4$

factored form:
solution(s):

1. $x^2 + 4x - 32$

factored form:
solution(s):

Explanation:

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

Step 1: Identify the form

The quadratic \( x^2 - 4x + 4 \) is a perfect square trinomial. The formula for a perfect square trinomial is \( a^2 - 2ab + b^2 = (a - b)^2 \). Here, \( a = x \) and \( 2ab = 4x \), so \( b = 2 \) (since \( 2 \times x \times 2 = 4x \)) and \( b^2 = 4 \).

Step 2: Factor the trinomial

Using the perfect square trinomial formula, we factor \( x^2 - 4x + 4 \) as \( (x - 2)^2 \).

Step 3: Find the solutions

To find the solutions, we set the factored form equal to zero: \( (x - 2)^2 = 0 \). Taking the square root of both sides, we get \( x - 2 = 0 \), so \( x = 2 \) (with multiplicity 2).

Factored Form: \( (x - 2)^2 \)

Solution(s): \( x = 2 \)

Problem 14: \( x^2 + 4x - 32 \)

Step 1: Find two numbers

We need two numbers that multiply to \( -32 \) (the constant term) and add up to \( 4 \) (the coefficient of the \( x \)-term). The numbers are \( 8 \) and \( -4 \) because \( 8 \times (-4) = -32 \) and \( 8 + (-4) = 4 \).

Step 2: Factor the quadratic

Using these two numbers, we can factor the quadratic as \( (x + 8)(x - 4) \) (since \( x^2 + 4x - 32 = x^2 + 8x - 4x - 32 = x(x + 8) - 4(x + 8) = (x + 8)(x - 4) \)).

Step 3: Find the solutions (optional, but if needed)

To find the solutions, set each factor equal to zero: \( x + 8 = 0 \) gives \( x = -8 \), and \( x - 4 = 0 \) gives \( x = 4 \).

Factored Form: \( (x + 8)(x - 4) \)

Solution(s): \( x = -8, x = 4 \)

Answer:

Step 1: Find two numbers

We need two numbers that multiply to \( -32 \) (the constant term) and add up to \( 4 \) (the coefficient of the \( x \)-term). The numbers are \( 8 \) and \( -4 \) because \( 8 \times (-4) = -32 \) and \( 8 + (-4) = 4 \).

Step 2: Factor the quadratic

Using these two numbers, we can factor the quadratic as \( (x + 8)(x - 4) \) (since \( x^2 + 4x - 32 = x^2 + 8x - 4x - 32 = x(x + 8) - 4(x + 8) = (x + 8)(x - 4) \)).

Step 3: Find the solutions (optional, but if needed)

To find the solutions, set each factor equal to zero: \( x + 8 = 0 \) gives \( x = -8 \), and \( x - 4 = 0 \) gives \( x = 4 \).

Factored Form: \( (x + 8)(x - 4) \)

Solution(s): \( x = -8, x = 4 \)