To determine which of the given equations can be solved using the quadratic formula, we need to first simplify each equation and see if it forms a quadratic equation. A quadratic equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
### Analyze Each Equation
1. Equation A: [tex]\( 2x^2 - 6x - 7 = 2x^2 \)[/tex]
- Subtract [tex]\( 2x^2 \)[/tex] from both sides to simplify:
[tex]\[
2x^2 - 6x - 7 - 2x^2 = 0
\][/tex]
[tex]\[
-6x - 7 = 0
\][/tex]
- This simplifies to a linear equation, not a quadratic equation.
2. Equation B: [tex]\( 2x^2 - 3x + 10 = 2x + 21 \)[/tex]
- Move all terms to one side to get:
[tex]\[
2x^2 - 3x + 10 - 2x - 21 = 0
\][/tex]
[tex]\[
2x^2 - 5x - 11 = 0
\][/tex]
- This is a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] with coefficients [tex]\( a = 2 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -11 \)[/tex].
3. Equation C: [tex]\( 5x^3 - 3x + 10 = 2x^2 \)[/tex]
- Move all terms to one side to get:
[tex]\[
5x^3 - 3x + 10 - 2x^2 = 0
\][/tex]
[tex]\[
5x^3 - 2x^2 - 3x + 10 = 0
\][/tex]
- This is a cubic equation due to the [tex]\( x^3 \)[/tex] term, and not a quadratic equation.
4. Equation D: [tex]\( 5x^2 + 2x - 4 = 2x^2 \)[/tex]
- Subtract [tex]\( 2x^2 \)[/tex] from both sides to simplify:
[tex]\[
5x^2 + 2x - 4 - 2x^2 = 0
\][/tex]
[tex]\[
3x^2 + 2x - 4 = 0
\][/tex]
- This is a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] with coefficients [tex]\( a = 3 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -4 \)[/tex].
### Conclusion
The two equations that can be solved using the quadratic formula are:
- B. [tex]\( 2x^2 - 3x + 10 = 2x + 21 \)[/tex] simplifies to [tex]\( 2x^2 - 5x - 11 = 0 \)[/tex]
- D. [tex]\( 5x^2 + 2x - 4 = 2x^2 \)[/tex] simplifies to [tex]\( 3x^2 + 2x - 4 = 0 \)[/tex]
Thus, the two equations that could be solved using the quadratic formula are B and D.
