Question

After being rearranged and simplified, which two of the following equations could be solved using the quadratic formula?
+2x2-3x+10=2x2+21
+5x3+2x-4=2x2
05x2-3x+10=2x2
0x2-6x-7=2x

Answer 1

To determine which equations can be solved using the quadratic formula, we need to rewrite them in standard quadratic form: \(ax^2 + bx + c = 0\). An equation can be solved using the quadratic formula if it includes an \(x^2\) term, and not more than a second-degree term.

Let's analyze each given equation:

1. **Equation**: \(2x^2 - 3x + 10 = 2x^2 + 21\)
- Rearranging: \(2x^2 - 2x^2 - 3x + 10 - 21 = 0\)
- Simplified: \(-3x - 11 = 0\) or \(3x + 11 = 0\), which is a linear equation (not quadratic).

2. **Equation**: \(5x^3 + 2x - 4 = 2x^2\)
- Rearranging: \(5x^3 - 2x^2 + 2x - 4 = 0\).
- This is a cubic equation (highest degree is 3, not 2).

3. **Equation**: \(5x^2 - 3x + 10 = 2x^2\)
- Rearranging: \(5x^2 - 2x^2 - 3x + 10 = 0\)
- Simplified: \(3x^2 - 3x + 10 = 0\).
- This is a quadratic equation.

4. **Equation**: \(0x^2 - 6x - 7 = 2x\)
- Rearranging: \(-6x - 7 - 2x = 0\)
- Simplified: \(-8x - 7 = 0\) or \(8x + 7 = 0\), which is a linear equation (not quadratic).

After analyzing the equations, we find that the only candidates for the quadratic formula are the third equation:

- **\(5x^2 - 3x + 10 = 2x^2\)** simplifies to \(3x^2 - 3x + 10 = 0\).

So out of the original options, only the third equation can be solved using the quadratic formula:

**Answer**: \(5x^2 - 3x + 10 = 2x^2\)