Answer :
Let's look at the solution step by step to find and correct Marsean's mistake in solving the quadratic equation.
1. Marsean started with the equation:
[tex]\[
25x^2 = 20x - 4
\][/tex]
2. He rearranged it to:
[tex]\[
25x^2 - 20x + 4 = 0
\][/tex]
Until this point, everything is correct.
3. He identified the coefficients as:
- [tex]\(a = 25\)[/tex]
- [tex]\(b = -20\)[/tex]
- [tex]\(c = 4\)[/tex]
4. He used the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Marsean substituted the values into the formula:
[tex]\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 25 \cdot 4}}{2 \cdot 25}
\][/tex]
6. At this step, Marsean calculated:
[tex]\[
x = \frac{20 \pm \sqrt{400 - 400}}{50}
\][/tex]
Here, the mistake was made in calculating the square root term. The discriminant (inside the square root) should be [tex]\(400 - 400\)[/tex], which equals [tex]\(0\)[/tex], not [tex]\(-800\)[/tex].
7. Now, correcting the calculation:
[tex]\[
x = \frac{20 \pm \sqrt{0}}{50}
\][/tex]
8. Since the square root of 0 is 0, this simplifies the solutions to:
[tex]\[
x = \frac{20}{50}
\][/tex]
9. Simplifying further yields:
[tex]\[
x = \frac{2}{5}
\][/tex]
This corrected calculation shows that the solution is [tex]\(x = \frac{2}{5}\)[/tex]. The mistake was in the calculation of the discriminant, originally noted in line 6. By fixing this, we confirmed that the correct value for [tex]\(x\)[/tex] is indeed [tex]\(\frac{2}{5}\)[/tex].
1. Marsean started with the equation:
[tex]\[
25x^2 = 20x - 4
\][/tex]
2. He rearranged it to:
[tex]\[
25x^2 - 20x + 4 = 0
\][/tex]
Until this point, everything is correct.
3. He identified the coefficients as:
- [tex]\(a = 25\)[/tex]
- [tex]\(b = -20\)[/tex]
- [tex]\(c = 4\)[/tex]
4. He used the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Marsean substituted the values into the formula:
[tex]\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 25 \cdot 4}}{2 \cdot 25}
\][/tex]
6. At this step, Marsean calculated:
[tex]\[
x = \frac{20 \pm \sqrt{400 - 400}}{50}
\][/tex]
Here, the mistake was made in calculating the square root term. The discriminant (inside the square root) should be [tex]\(400 - 400\)[/tex], which equals [tex]\(0\)[/tex], not [tex]\(-800\)[/tex].
7. Now, correcting the calculation:
[tex]\[
x = \frac{20 \pm \sqrt{0}}{50}
\][/tex]
8. Since the square root of 0 is 0, this simplifies the solutions to:
[tex]\[
x = \frac{20}{50}
\][/tex]
9. Simplifying further yields:
[tex]\[
x = \frac{2}{5}
\][/tex]
This corrected calculation shows that the solution is [tex]\(x = \frac{2}{5}\)[/tex]. The mistake was in the calculation of the discriminant, originally noted in line 6. By fixing this, we confirmed that the correct value for [tex]\(x\)[/tex] is indeed [tex]\(\frac{2}{5}\)[/tex].
