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? enter this equation as a quadratic equation in standard form $ax^2 + bx + c = 0$

Explanation:

Step1: Translate the problem into an equation

Let the positive number be \( x \). Four more than the number is \( x + 4 \). The product of the number and four more than the number is \( x(x + 4) \), and this product is equal to 32. So we have the equation:
\( x(x + 4)=32 \)

Step2: Expand and rearrange into standard quadratic form

First, expand the left - hand side: \( x(x + 4)=x^{2}+4x \).
Then, subtract 32 from both sides to get the equation in the form \( ax^{2}+bx + c = 0 \):
\( x^{2}+4x-32 = 0 \)

Step3: Solve the quadratic equation \( x^{2}+4x - 32=0 \)

We can factor the quadratic equation. We need two numbers that multiply to \( - 32 \) and add up to 4. The numbers are 8 and - 4.
So, \( x^{2}+4x - 32=(x + 8)(x - 4)=0 \)
Setting each factor equal to zero:
\( x+8 = 0\) or \(x - 4=0\)
\( x=-8\) or \(x = 4\)
Since the number is positive, we discard \( x=-8 \).

Answer:

The lower (positive) number is \( 4 \), and the quadratic equation in standard form is \( x^{2}+4x - 32=0 \)