We appreciate your visit to 13 Marsean solved the following quadratic equation but did not end up with the correct answer Find his mistake circle the line where the mistake. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
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