Answer :
Let's go through Marsean's solution step-by-step to identify and correct the mistake:
1. Original Equation:
[tex]\( 25x^2 = 20x - 4 \)[/tex]
2. Rearrange into Standard Quadratic Form:
[tex]\( 25x^2 - 20x + 4 = 0 \)[/tex]
3. Coefficients:
[tex]\( a = 25 \)[/tex], [tex]\( b = -20 \)[/tex], [tex]\( c = 4 \)[/tex]
4. Quadratic Formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Discriminant Calculation:
Let's calculate the discriminant, which is the part under the square root:
[tex]\( b^2 - 4ac = (-20)^2 - 4 \cdot 25 \cdot 4 \)[/tex]
6. Calculate Discriminant:
[tex]\( = 400 - 400 = 0 \)[/tex]
Here is where Marsean made a mistake. The discriminant was calculated as [tex]\(-800\)[/tex] in his work, which was incorrect.
7. Square Root of the Discriminant:
Since the corrected discriminant is 0:
[tex]\(\sqrt{0} = 0\)[/tex]
8. Plug Back into the Quadratic Formula:
[tex]\[
x = \frac{-(-20) \pm 0}{2 \cdot 25} = \frac{20 \pm 0}{50}
\][/tex]
9. Simplify the Expression:
[tex]\[
x = \frac{20}{50} = \frac{2}{5}
\][/tex]
The correct solution shows that there is a single real solution, [tex]\( x = \frac{2}{5} \)[/tex], not a complex one as originally stated. Therefore, the mistake was in calculating the discriminant incorrectly as [tex]\(-800\)[/tex] instead of [tex]\(0\)[/tex], and consequently finding complex solutions instead of a real solution.
1. Original Equation:
[tex]\( 25x^2 = 20x - 4 \)[/tex]
2. Rearrange into Standard Quadratic Form:
[tex]\( 25x^2 - 20x + 4 = 0 \)[/tex]
3. Coefficients:
[tex]\( a = 25 \)[/tex], [tex]\( b = -20 \)[/tex], [tex]\( c = 4 \)[/tex]
4. Quadratic Formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Discriminant Calculation:
Let's calculate the discriminant, which is the part under the square root:
[tex]\( b^2 - 4ac = (-20)^2 - 4 \cdot 25 \cdot 4 \)[/tex]
6. Calculate Discriminant:
[tex]\( = 400 - 400 = 0 \)[/tex]
Here is where Marsean made a mistake. The discriminant was calculated as [tex]\(-800\)[/tex] in his work, which was incorrect.
7. Square Root of the Discriminant:
Since the corrected discriminant is 0:
[tex]\(\sqrt{0} = 0\)[/tex]
8. Plug Back into the Quadratic Formula:
[tex]\[
x = \frac{-(-20) \pm 0}{2 \cdot 25} = \frac{20 \pm 0}{50}
\][/tex]
9. Simplify the Expression:
[tex]\[
x = \frac{20}{50} = \frac{2}{5}
\][/tex]
The correct solution shows that there is a single real solution, [tex]\( x = \frac{2}{5} \)[/tex], not a complex one as originally stated. Therefore, the mistake was in calculating the discriminant incorrectly as [tex]\(-800\)[/tex] instead of [tex]\(0\)[/tex], and consequently finding complex solutions instead of a real solution.
