Sure, let's analyze each equation one by one to determine if it can be simplified to a quadratic equation, and then see which ones can be solved using the quadratic formula.
1. Equation A: [tex]\(5 x^3 - 3 x + 10 = 2 x^2\)[/tex]
Rearrange and simplify:
[tex]\[
5 x^3 - 3 x + 10 - 2 x^2 = 0
\][/tex]
[tex]\[
5 x^3 - 2 x^2 - 3 x + 10 = 0
\][/tex]
This is a cubic equation (degree 3), so it cannot be solved using the quadratic formula.
2. Equation B: [tex]\(2 x^2 - 3 x + 10 = 2 x + 21\)[/tex]
Rearrange and simplify:
[tex]\[
2 x^2 - 3 x + 10 - 2 x - 21 = 0
\][/tex]
[tex]\[
2 x^2 - 5 x - 11 = 0
\][/tex]
This is a quadratic equation (degree 2), so it can be solved using the quadratic formula.
3. Equation C: [tex]\(5 x^2 + 2 x - 4 = 2 x^2\)[/tex]
Rearrange and simplify:
[tex]\[
5 x^2 + 2 x - 4 - 2 x^2 = 0
\][/tex]
[tex]\[
3 x^2 + 2 x - 4 = 0
\][/tex]
This is a quadratic equation (degree 2), so it can be solved using the quadratic formula.
4. Equation D: [tex]\(2 x^2 - 6 x - 7 = 2 x^2\)[/tex]
Rearrange and simplify:
[tex]\[
2 x^2 - 6 x - 7 - 2 x^2 = 0
\][/tex]
[tex]\[
-6 x - 7 = 0
\][/tex]
[tex]\[
-6 x = 7
\][/tex]
[tex]\[
x = -\frac{7}{6}
\][/tex]
This is a linear equation (degree 1), so it cannot be solved using the quadratic formula.
So, the equations that can be solved using the quadratic formula are:
B. [tex]\(2 x^2 - 3 x + 10 = 2 x + 21\)[/tex]
C. [tex]\(5 x^2 + 2 x - 4 = 2 x^2\)[/tex]
