To determine the solutions to the given equation [tex]\(x^2 - 6x + 40 = 6x + 5\)[/tex], we need to solve for [tex]\(x\)[/tex]. Here's a detailed, step-by-step solution:
1. Rewrite the equation such that all terms are on one side:
[tex]\[
x^2 - 6x + 40 = 6x + 5
\][/tex]
Subtract [tex]\(6x\)[/tex] and 5 from both sides:
[tex]\[
x^2 - 6x + 40 - 6x - 5 = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 - 12x + 35 = 0
\][/tex]
2. Factor the quadratic equation [tex]\(x^2 - 12x + 35 = 0\)[/tex]:
We look for two numbers that multiply to 35 and add up to -12. These numbers are -7 and -5.
So, we can factor the equation as:
[tex]\[
(x - 7)(x - 5) = 0
\][/tex]
3. Solve for [tex]\(x\)[/tex] using the factored form:
Set each factor equal to zero:
[tex]\[
x - 7 = 0 \quad \text{or} \quad x - 5 = 0
\][/tex]
Solve each equation:
[tex]\[
x = 7 \quad \text{or} \quad x = 5
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 7\)[/tex] and [tex]\(x = 5\)[/tex].
4. Check the given options:
A. [tex]\(7\)[/tex] -- This is a solution.
B. [tex]\(-5\)[/tex] -- This is not a solution.
C. [tex]\(6\)[/tex] -- This is not a solution.
D. [tex]\(5\)[/tex] -- This is a solution.
E. [tex]\(-6\)[/tex] -- This is not a solution.
Therefore, the correct options that are solutions to the equation [tex]\(x^2 - 6x + 40 = 6x + 5\)[/tex] are:
- A. 7
- D. 5
