Question

the product of a positive number and four more than the number is 32. use x for the lower number. what is the lower number? equation?

Explanation:

Step1: Formulate the equation

Let the positive number be \( x \). Four more than the number is \( x + 4 \). The product of \( x \) and \( x + 4 \) is 32, so the equation is \( x(x + 4)=32 \).

Step2: Expand and simplify the equation

Expand the left - hand side: \( x^{2}+4x = 32 \).
Rearrange it to the standard quadratic form \( ax^{2}+bx + c = 0 \): \( x^{2}+4x-32 = 0 \).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to - 32 and add up to 4. The numbers are 8 and - 4.
So, \( x^{2}+4x - 32=(x + 8)(x - 4)=0 \).

Step4: Solve for \( x \)

Set each factor equal to zero:

• If \( x+8 = 0 \), then \( x=-8 \). But we are looking for a positive number, so we discard this solution.
• If \( x - 4=0 \), then \( x = 4 \).

Answer:

The lower number is 4.