To determine which equations could be solved using the quadratic formula, we need to rearrange and simplify each equation. The quadratic formula can be used to solve equations of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Let's check each equation step-by-step:
Equation A: [tex]\(2x^2 - 3x + 10 = 2x^2 + 21\)[/tex]
1. Subtract [tex]\(2x^2 + 21\)[/tex] from both sides to bring it to standard quadratic form:
[tex]\[
2x^2 - 3x + 10 - (2x^2 + 21) = 0
\][/tex]
2. Simplify:
[tex]\[
2x^2 - 3x + 10 - 2x^2 - 21 = 0
\][/tex]
[tex]\[
-3x - 11 = 0
\][/tex]
After simplification, this equation is not quadratic as it does not have an [tex]\( x^2 \)[/tex] term.
Equation B: [tex]\(5x^3 + 2x - 4 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[
5x^3 + 2x - 4 - 2x^2 = 0
\][/tex]
2. Simplify:
[tex]\[
5x^3 - 2x^2 + 2x - 4 = 0
\][/tex]
This is a cubic equation (degree 3), thus not a quadratic equation.
Equation C: [tex]\(5x^2 - 3x + 10 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[
5x^2 - 3x + 10 - 2x^2 = 0
\][/tex]
2. Simplify:
[tex]\[
3x^2 - 3x + 10 = 0
\][/tex]
This is a quadratic equation.
Equation D: [tex]\(x^2 - 6x - 7 = 2x\)[/tex]
1. Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
x^2 - 6x - 7 - 2x = 0
\][/tex]
2. Simplify:
[tex]\[
x^2 - 8x - 7 = 0
\][/tex]
This is also a quadratic equation.
Therefore, after rearranging and simplifying, the equations that can be solved using the quadratic formula are:
- C: [tex]\(3x^2 - 3x + 10 = 0\)[/tex]
- D: [tex]\(x^2 - 8x - 7 = 0\)[/tex]
Thus, the correct answers are:
[tex]\[
\boxed{\text{C and D}}
\][/tex]
