High School

Marsean solved the following quadratic equation. He didn't end up with the correct answer. Find his mistake, circle the line where the mistake was made, and correct the mistake to get the correct answer of [tex]$x=\frac{2}{5}$[/tex].

\[
\begin{array}{l}
1. \quad 25x^2 = 20x - 4 \\
2. \quad 25x^2 - 20x + 4 = 0 \\
3. \quad a = 25, \, b = -20, \, c = 4 \\
4. \quad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\
5. \quad x = \frac{--20 \pm \sqrt{-20^2 - 4(25)(4)}}{2(25)} \\
6. \quad x = \frac{20 \pm \sqrt{400 - 400}}{50} \\
7. \quad x = \frac{20 \pm \sqrt{0}}{50} \\
8. \quad x = \frac{20}{50} \\
9. \quad x = \frac{2}{5}
\end{array}
\]

**Mistake Found in Line 5:**

- Incorrect expression: [tex]x = \frac{--20 \pm \sqrt{-20^2 - 4(25)(4)}}{2(25)}[/tex]
- Correct expression: [tex]x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(25)(4)}}{2(25)}[/tex]

**Correct Solution:**

The solution [tex]x = \frac{2}{5}[/tex] is real and correct.

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].