Answer :
To identify and correct Marsean's mistake in solving the quadratic equation, let's go through the solution step-by-step:
Original Equation:
[tex]\[ 25x^2 = 20x - 4 \][/tex]
Step 1: Rewriting the Equation in Standard Form
To get the equation into standard form, we need to move all terms to one side:
[tex]\[ 25x^2 - 20x + 4 = 0 \][/tex]
Step 2: Identifying Coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. From our equation:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = -20 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Step 3: Quadratic Formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step 4: Calculating the Discriminant
The discriminant is the part under the square root in the quadratic formula:
[tex]\[ b^2 - 4ac \][/tex]
Calculate this term:
[tex]\[ b^2 = (-20)^2 = 400 \][/tex]
[tex]\[ 4ac = 4 \times 25 \times 4 = 400 \][/tex]
Thus, the discriminant is:
[tex]\[ 400 - 400 = 0 \][/tex]
Step 5: Solving for x using the Quadratic Formula
Since the discriminant is 0, this means there is one real and repeated root:
[tex]\[ x = \frac{-b \pm \sqrt{0}}{2a} \][/tex]
Simplify:
[tex]\[ x = \frac{20}{50} = \frac{2}{5} \][/tex]
Mistake Identification:
Marsean made a mistake when calculating the term inside the square root (the discriminant). He should have calculated the discriminant as [tex]\( 0 \)[/tex] instead of a negative value, which led to his incorrect statement of "No real solution."
Correct Solution:
The correct and repeated solution to the quadratic equation is:
[tex]\[ x = \frac{2}{5} \][/tex]
This step-by-step approach shows how the error occurred and how to solve the equation correctly.
Original Equation:
[tex]\[ 25x^2 = 20x - 4 \][/tex]
Step 1: Rewriting the Equation in Standard Form
To get the equation into standard form, we need to move all terms to one side:
[tex]\[ 25x^2 - 20x + 4 = 0 \][/tex]
Step 2: Identifying Coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. From our equation:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = -20 \)[/tex]
- [tex]\( c = 4 \)[/tex]
Step 3: Quadratic Formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step 4: Calculating the Discriminant
The discriminant is the part under the square root in the quadratic formula:
[tex]\[ b^2 - 4ac \][/tex]
Calculate this term:
[tex]\[ b^2 = (-20)^2 = 400 \][/tex]
[tex]\[ 4ac = 4 \times 25 \times 4 = 400 \][/tex]
Thus, the discriminant is:
[tex]\[ 400 - 400 = 0 \][/tex]
Step 5: Solving for x using the Quadratic Formula
Since the discriminant is 0, this means there is one real and repeated root:
[tex]\[ x = \frac{-b \pm \sqrt{0}}{2a} \][/tex]
Simplify:
[tex]\[ x = \frac{20}{50} = \frac{2}{5} \][/tex]
Mistake Identification:
Marsean made a mistake when calculating the term inside the square root (the discriminant). He should have calculated the discriminant as [tex]\( 0 \)[/tex] instead of a negative value, which led to his incorrect statement of "No real solution."
Correct Solution:
The correct and repeated solution to the quadratic equation is:
[tex]\[ x = \frac{2}{5} \][/tex]
This step-by-step approach shows how the error occurred and how to solve the equation correctly.
