Both Michaela's and Grace's solution strategies work. Let's go through each strategy step-by-step to verify this.
### Michaela's Strategy:
1. Equation Setup:
[tex]\( 4x^2 - 100 = 0 \)[/tex]
2. Factoring:
Recognize that the equation can be factored as the difference of squares.
[tex]\[
4x^2 - 100 = (2x)^2 - 10^2 = (2x + 10)(2x - 10) = 0
\][/tex]
3. Zero Product Property:
Apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
[tex]\[
2x + 10 = 0 \quad \text{or} \quad 2x - 10 = 0
\][/tex]
4. Solving Each Equation:
- For [tex]\( 2x + 10 = 0 \)[/tex]:
[tex]\[
2x = -10 \implies x = -5
\][/tex]
- For [tex]\( 2x - 10 = 0 \)[/tex]:
[tex]\[
2x = 10 \implies x = 5
\][/tex]
5. Solutions:
[tex]\[
x = -5 \quad \text{or} \quad x = 5
\][/tex]
### Grace's Strategy:
1. Equation Setup:
[tex]\( 4x^2 - 100 = 0 \)[/tex]
2. Isolating [tex]\( x^2 \)[/tex]:
Add 100 to both sides to isolate the [tex]\(x^2\)[/tex] term.
[tex]\[
4x^2 = 100
\][/tex]
3. Dividing by 4:
Divide both sides by 4 to further isolate [tex]\(x^2\)[/tex].
[tex]\[
x^2 = 25
\][/tex]
4. Taking the Square Root:
Take the square root of both sides to solve for [tex]\( x \)[/tex], remembering to consider both positive and negative roots.
[tex]\[
x = \pm \sqrt{25}
\][/tex]
[tex]\[
x = \pm 5
\][/tex]
5. Solutions:
[tex]\[
x = -5 \quad \text{or} \quad x = 5
\][/tex]
### Conclusion:
Both strategies correctly solve the equation [tex]\( 4x^2 - 100 = 0 \)[/tex]. Michaela's factoring method and Grace's isolating and square root method both yield the solutions:
[tex]\[
x = -5 \quad \text{or} \quad x = 5
\][/tex]
Thus, both Michaela's and Grace's solution strategies work.
