To find out which equations can be solved using the quadratic formula, we need to identify which of the given equations are quadratic equations. A quadratic equation is any equation that can be written in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex].
Let's analyze each equation one by one:
### Equation A: [tex]\(2x^2 - 6x - 7 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides to simplify:
[tex]\[
2x^2 - 6x - 7 - 2x^2 = 2x^2 - 2x^2
\][/tex]
2. Simplify:
[tex]\[
-6x - 7 = 0
\][/tex]
This is a linear equation, not a quadratic equation because it does not have an [tex]\(x^2\)[/tex] term.
### Equation B: [tex]\(2x^2 - 3x + 10 = 2x + 21\)[/tex]
1. Subtract [tex]\(2x + 21\)[/tex] from both sides:
[tex]\[
2x^2 - 3x + 10 - 2x - 21 = 0
\][/tex]
2. Combine like terms:
[tex]\[
2x^2 - 5x - 11 = 0
\][/tex]
This is a quadratic equation because it is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 2\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -11\)[/tex].
### Equation C: [tex]\(5x^3 - 3x + 10 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[
5x^3 - 3x + 10 - 2x^2 = 0
\][/tex]
This is not a quadratic equation because it has an [tex]\(x^3\)[/tex] term. It is a cubic equation.
### Equation D: [tex]\(5x^2 + 2x - 4 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[
5x^2 + 2x - 4 - 2x^2 = 0
\][/tex]
2. Simplify:
[tex]\[
3x^2 + 2x - 4 = 0
\][/tex]
This is a quadratic equation because it is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 3\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -4\)[/tex].
### Conclusion
The equations that can be solved using the quadratic formula are:
- Equation B: [tex]\(2x^2 - 5x - 11 = 0\)[/tex]
- Equation D: [tex]\(3x^2 + 2x - 4 = 0\)[/tex]
Therefore, the two equations that could be solved using the quadratic formula are B and D.
