To determine which of the given equations could be solved using the quadratic formula, we need to identify whether their simplified forms are quadratic equations. A quadratic equation is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Let's consider each equation individually:
### Equation A:
[tex]\[ 2x^2 - 3x + 10 = 2x^2 + 21 \][/tex]
1. Subtract [tex]\( 2x^2 + 21 \)[/tex] from both sides:
[tex]\[ 2x^2 - 3x + 10 - (2x^2 + 21) = 0 \][/tex]
2. Simplify the equation:
[tex]\[ 2x^2 - 3x + 10 - 2x^2 - 21 = 0 \][/tex]
[tex]\[ -3x - 11 = 0 \][/tex]
3. The simplified form is:
[tex]\[ -3x = 11 \][/tex]
This is a linear equation, not quadratic.
### Equation B:
[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 the equation:
[tex]\[ 3x^2 - 3x + 10 = 0 \][/tex]
This is a quadratic equation.
### Equation C:
[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]
This is a cubic equation due to the presence of [tex]\( x^3 \)[/tex], which is not quadratic.
### 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 the equation:
[tex]\[ x^2 - 8x - 7 = 0 \][/tex]
This is a quadratic equation.
In conclusion, the equations that could be solved using the quadratic formula are:
[tex]\[ \text{Equation B: } 5x^2 - 3x + 10 = 2x^2 \][/tex]
[tex]\[ \text{Equation D: } x^2 - 6x - 7 = 2x \][/tex]
Therefore, the correct options are:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
