Answer :
To determine which rearranged and simplified equation cannot be solved using the quadratic formula, let’s analyze each equation step by step.
### Equation A: [tex]\(5x^2 - 3x + 10 = 5x^2\)[/tex]
1. Subtract [tex]\(5x^2\)[/tex] from both sides:
[tex]\[ 5x^2 - 3x + 10 - 5x^2 = 0 \][/tex]
2. Simplify:
[tex]\[ -3x + 10 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ -3x + 10 = 0 \][/tex]
This is a linear equation, not a quadratic equation.
### Equation B: [tex]\(5x^2 - 3x + 10 = 2x^2 + 21\)[/tex]
1. Subtract [tex]\(2x^2 + 21\)[/tex] from both sides:
[tex]\[ 5x^2 - 3x + 10 - 2x^2 - 21 = 0 \][/tex]
2. Simplify:
[tex]\[ 3x^2 - 3x - 11 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ 3x^2 - 3x - 11 = 0 \][/tex]
This is a quadratic equation.
### Equation C: [tex]\(2x - 4 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[ 2x - 4 - 2x^2 = 0 \][/tex]
2. Simplify:
[tex]\[ -2x^2 + 2x - 4 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ -2x^2 + 2x - 4 = 0 \][/tex]
This is a quadratic equation.
### Equation D: [tex]\(x^2 - 6x - 7 = 2\)[/tex]
1. Subtract 2 from both sides:
[tex]\[ x^2 - 6x - 7 - 2 = 0 \][/tex]
2. Simplify:
[tex]\[ x^2 - 6x - 9 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ x^2 - 6x - 9 = 0 \][/tex]
This is a quadratic equation.
### Conclusion:
The only equation that simplifies to a linear equation (and not a quadratic equation) is:
[tex]\[ 5x^2 - 3x + 10 = 5x^2 \][/tex]
Thus, the equation that cannot be solved using the quadratic formula is:
Equation A: [tex]\(5x^2 - 3x + 10 = 5x^2\)[/tex]
### Equation A: [tex]\(5x^2 - 3x + 10 = 5x^2\)[/tex]
1. Subtract [tex]\(5x^2\)[/tex] from both sides:
[tex]\[ 5x^2 - 3x + 10 - 5x^2 = 0 \][/tex]
2. Simplify:
[tex]\[ -3x + 10 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ -3x + 10 = 0 \][/tex]
This is a linear equation, not a quadratic equation.
### Equation B: [tex]\(5x^2 - 3x + 10 = 2x^2 + 21\)[/tex]
1. Subtract [tex]\(2x^2 + 21\)[/tex] from both sides:
[tex]\[ 5x^2 - 3x + 10 - 2x^2 - 21 = 0 \][/tex]
2. Simplify:
[tex]\[ 3x^2 - 3x - 11 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ 3x^2 - 3x - 11 = 0 \][/tex]
This is a quadratic equation.
### Equation C: [tex]\(2x - 4 = 2x^2\)[/tex]
1. Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[ 2x - 4 - 2x^2 = 0 \][/tex]
2. Simplify:
[tex]\[ -2x^2 + 2x - 4 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ -2x^2 + 2x - 4 = 0 \][/tex]
This is a quadratic equation.
### Equation D: [tex]\(x^2 - 6x - 7 = 2\)[/tex]
1. Subtract 2 from both sides:
[tex]\[ x^2 - 6x - 7 - 2 = 0 \][/tex]
2. Simplify:
[tex]\[ x^2 - 6x - 9 = 0 \][/tex]
3. The simplified equation is:
[tex]\[ x^2 - 6x - 9 = 0 \][/tex]
This is a quadratic equation.
### Conclusion:
The only equation that simplifies to a linear equation (and not a quadratic equation) is:
[tex]\[ 5x^2 - 3x + 10 = 5x^2 \][/tex]
Thus, the equation that cannot be solved using the quadratic formula is:
Equation A: [tex]\(5x^2 - 3x + 10 = 5x^2\)[/tex]
