Question

question
the difference of the square of a number and 9 is equal to 8 times that number. find the positive solution.

Explanation:

Step1: Define the variable

Let the number be \( x \). The square of the number is \( x^2 \), the difference of the square of the number and 9 is \( x^2 - 9 \), and 8 times the number is \( 8x \). So we can set up the equation: \( x^2 - 9 = 8x \).

Step2: Rearrange the equation

Rearrange the equation to the standard quadratic form \( ax^2 + bx + c = 0 \). Subtract \( 8x \) from both sides: \( x^2 - 8x - 9 = 0 \).

Step3: Factor the quadratic equation

Factor the quadratic equation. We need two numbers that multiply to -9 and add to -8. The numbers are -9 and 1. So, \( (x - 9)(x + 1) = 0 \).

Step4: Solve for \( x \)

Set each factor equal to zero:

• \( x - 9 = 0 \) gives \( x = 9 \)
• \( x + 1 = 0 \) gives \( x = -1 \)

Step5: Find the positive solution

We need the positive solution, so from the solutions \( x = 9 \) and \( x = -1 \), the positive solution is \( x = 9 \).

Answer:

\( 9 \)