Answer :
To determine which of the given choices are solutions to the equation [tex]\(x^2 + 6x + 9 = 20\)[/tex], we need to simplify this equation first by moving all terms to one side:
[tex]\[ x^2 + 6x + 9 - 20 = 0 \][/tex]
[tex]\[ x^2 + 6x - 11 = 0 \][/tex]
This simplified form is what we'll use to evaluate each of the provided choices:
A. [tex]\( x = -2 \sqrt{5} + 3 \)[/tex]
[tex]\[ \left( -2 \sqrt{5} + 3 \right)^2 + 6 \left( -2 \sqrt{5} + 3 \right) - 11 = 0 \\ \][/tex]
Evaluating it does not satisfy [tex]\(x^2 + 6x - 11 = 0\)[/tex].
B. [tex]\( x = -2 \sqrt{5} - 3 \)[/tex]
[tex]\[ \left( -2 \sqrt{5} - 3 \right)^2 + 6 \left( -2 \sqrt{5} - 3 \right) - 11 = 0 \\ \][/tex]
Evaluation shows it does not satisfy the equation.
C. [tex]\( x = 2 \sqrt{5} - 3 \)[/tex]
[tex]\[ \left( 2 \sqrt{5} - 3 \right)^2 + 6 \left( 2 \sqrt{5} - 3 \right) - 11 = 0 \\ \][/tex]
This choice does not satisfy the equation.
D. [tex]\( x = -\sqrt{3} + 28 \)[/tex]
[tex]\[ \left( -\sqrt{3} + 28 \right)^2 + 6 \left( -\sqrt{3} + 28 \right) - 11 = 0 \\ \][/tex]
This does not satisfy the equation.
E. [tex]\( x = 2 \sqrt{5} + 3 \)[/tex]
[tex]\[ \left( 2 \sqrt{5} + 3 \right)^2 + 6 \left( 2 \sqrt{5} + 3 \right) - 11 = 0 \\ \][/tex]
This choice does not satisfy the equation.
F. [tex]\( x = \sqrt{3} + 20 \)[/tex]
[tex]\[ \left( \sqrt{3} + 20 \right)^2 + 6 \left( \sqrt{3} + 20 \right) - 11 = 0 \\ \][/tex]
This does not satisfy the equation.
After checking all provided solutions, none of the choices (A, B, C, D, E, F) satisfy the equation [tex]\(x^2 + 6x - 11 = 0\)[/tex].
Thus, the correct answer is:
None of the provided choices (A, B, C, D, E, F) are solutions to the equation [tex]\(x^2 + 6x + 9 = 20\)[/tex].
[tex]\[ x^2 + 6x + 9 - 20 = 0 \][/tex]
[tex]\[ x^2 + 6x - 11 = 0 \][/tex]
This simplified form is what we'll use to evaluate each of the provided choices:
A. [tex]\( x = -2 \sqrt{5} + 3 \)[/tex]
[tex]\[ \left( -2 \sqrt{5} + 3 \right)^2 + 6 \left( -2 \sqrt{5} + 3 \right) - 11 = 0 \\ \][/tex]
Evaluating it does not satisfy [tex]\(x^2 + 6x - 11 = 0\)[/tex].
B. [tex]\( x = -2 \sqrt{5} - 3 \)[/tex]
[tex]\[ \left( -2 \sqrt{5} - 3 \right)^2 + 6 \left( -2 \sqrt{5} - 3 \right) - 11 = 0 \\ \][/tex]
Evaluation shows it does not satisfy the equation.
C. [tex]\( x = 2 \sqrt{5} - 3 \)[/tex]
[tex]\[ \left( 2 \sqrt{5} - 3 \right)^2 + 6 \left( 2 \sqrt{5} - 3 \right) - 11 = 0 \\ \][/tex]
This choice does not satisfy the equation.
D. [tex]\( x = -\sqrt{3} + 28 \)[/tex]
[tex]\[ \left( -\sqrt{3} + 28 \right)^2 + 6 \left( -\sqrt{3} + 28 \right) - 11 = 0 \\ \][/tex]
This does not satisfy the equation.
E. [tex]\( x = 2 \sqrt{5} + 3 \)[/tex]
[tex]\[ \left( 2 \sqrt{5} + 3 \right)^2 + 6 \left( 2 \sqrt{5} + 3 \right) - 11 = 0 \\ \][/tex]
This choice does not satisfy the equation.
F. [tex]\( x = \sqrt{3} + 20 \)[/tex]
[tex]\[ \left( \sqrt{3} + 20 \right)^2 + 6 \left( \sqrt{3} + 20 \right) - 11 = 0 \\ \][/tex]
This does not satisfy the equation.
After checking all provided solutions, none of the choices (A, B, C, D, E, F) satisfy the equation [tex]\(x^2 + 6x - 11 = 0\)[/tex].
Thus, the correct answer is:
None of the provided choices (A, B, C, D, E, F) are solutions to the equation [tex]\(x^2 + 6x + 9 = 20\)[/tex].
