Q. For what values of \(x\) is \(x^2 – 36 = 5x\) true?

Problem

Solve for x in the equation

\[x^2 – 36 = 5x\]

Step-by-step solution

1. Move all terms to one side to write the equation in standard quadratic form. Subtract 5x from both sides to obtain

   \[x^2 – 5x – 36 = 0\]

   Explanation: A quadratic equation is easiest to solve when it equals zero, so we combine like terms and collect everything on one side.

2. Factor the quadratic trinomial. We look for two numbers whose product is -36 and whose sum is -5. Those numbers are -9 and 4, because

   \[-9 \times 4 = -36 \quad\text{and}\quad -9 + 4 = -5\]

   So we can factor the quadratic as

   \[x^2 – 5x – 36 = (x – 9)(x + 4)\]

   Explanation: Factoring rewrites the quadratic as a product of two linear factors. Finding numbers that multiply to the constant term and add to the linear coefficient is the standard factoring technique for trinomials.

3. Use the zero-product property: if a product of two factors equals zero, then at least one factor must be zero. Therefore, set each factor equal to zero:

   \[x – 9 = 0 \quad\text{or}\quad x + 4 = 0\]

   Solving these gives

   \[x = 9 \quad\text{or}\quad x = -4\]

   Explanation: This gives the candidate solutions of the original equation.

4. Check each solution by substituting back into the original equation \[x^2 – 36 = 5x\].

   • For x = 9:

     \[9^2 – 36 = 81 – 36 = 45\]

     \[5 \cdot 9 = 45\]

     Both sides equal 45, so x = 9 is valid.

   • For x = -4:

     \[(-4)^2 – 36 = 16 – 36 = -20\]

     \[5 \cdot (-4) = -20\]

     Both sides equal -20, so x = -4 is valid.

   Explanation: Verifying solutions ensures no arithmetic or algebraic mistake occurred during solving.

Final answer

The equation \[x^2 – 36 = 5x\] is true for

x = 9 and x = -4.