To find the solutions to the equation [tex]\( x^2 - 3x + 27 = 6x + 7 \)[/tex], we first need to set the equation to zero by moving all terms to one side. Let’s simplify the equation step-by-step:
Starting with the original equation:
[tex]\[ x^2 - 3x + 27 = 6x + 7 \][/tex]
Subtract [tex]\( 6x \)[/tex] and [tex]\( 7 \)[/tex] from both sides to set the equation to zero:
[tex]\[ x^2 - 3x + 27 - 6x - 7 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 - 3x - 6x + 27 - 7 = 0 \][/tex]
[tex]\[ x^2 - 9x + 20 = 0 \][/tex]
Now we need to solve the quadratic equation [tex]\( x^2 - 9x + 20 = 0 \)[/tex]. We do this by factoring, completing the square, or using the quadratic formula. Factoring this equation, we look for two numbers that multiply to 20 and add up to -9.
The factors of 20 that add up to -9 are -4 and -5:
[tex]\[ x^2 - 9x + 20 = (x - 4)(x - 5) = 0 \][/tex]
Setting each factor to zero gives us:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
So the solutions to the equation [tex]\( x^2 - 9x + 20 = 0 \)[/tex] are [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex].
Now we need to check which of the provided options match these solutions. The given options are:
A. 3
B. 6
C. -4
D. 4
E. 5
Comparing these options with our solutions, [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex]:
- Option A: 3 — Not a solution
- Option B: 6 — Not a solution
- Option C: -4 — Not a solution
- Option D: 4 — Is a solution
- Option E: 5 — Is a solution
Therefore, the correct answers are D. 4 and E. 5.
