Answer :
To solve the given quadratic equation [tex]\(x^2 + 8x - 9 = 0\)[/tex] using the quadratic formula, we can follow these detailed steps:
1. Identify the coefficients:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = 8\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -9\)[/tex] (constant term)
2. Write the quadratic formula:
The quadratic formula is [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
3. Substitute the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ x = \frac{-(8) \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)} \][/tex]
4. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 8^2 - 4(1)(-9) = 64 + 36 = 100 \][/tex]
5. Calculate the square root of the discriminant:
[tex]\[ \sqrt{100} = 10 \][/tex]
6. Substitute back into the quadratic formula and solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-8 \pm 10}{2} \][/tex]
7. Find the two possible solutions:
- For [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-8 + 10}{2} = \frac{2}{2} = 1 \][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-8 - 10}{2} = \frac{-18}{2} = -9 \][/tex]
So the correct solutions should be [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
Now, let's examine Soren's steps to identify the error:
- Step 1: [tex]\(x = \frac{-8 \pm \sqrt{64 + 36}}{2}\)[/tex]
This step is correct, as we calculated the discriminant correctly (64 + 36 = 100).
- Step 2: [tex]\(x = \frac{-8 \pm \sqrt{100}}{2}\)[/tex]
This step is also correct, as [tex]\(\sqrt{100} = 10\)[/tex].
- Step 3: [tex]\(x = \{-18, 2\}\)[/tex]
Here is where Soren made an error. The correct computation using the formula should result in the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
Therefore, the error was made in Step 3. Soren did not properly solve the final expressions resulting from [tex]\(\frac{-8 \pm 10}{2}\)[/tex]. The correct results should have been [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
1. Identify the coefficients:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = 8\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -9\)[/tex] (constant term)
2. Write the quadratic formula:
The quadratic formula is [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
3. Substitute the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ x = \frac{-(8) \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)} \][/tex]
4. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 8^2 - 4(1)(-9) = 64 + 36 = 100 \][/tex]
5. Calculate the square root of the discriminant:
[tex]\[ \sqrt{100} = 10 \][/tex]
6. Substitute back into the quadratic formula and solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-8 \pm 10}{2} \][/tex]
7. Find the two possible solutions:
- For [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-8 + 10}{2} = \frac{2}{2} = 1 \][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-8 - 10}{2} = \frac{-18}{2} = -9 \][/tex]
So the correct solutions should be [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
Now, let's examine Soren's steps to identify the error:
- Step 1: [tex]\(x = \frac{-8 \pm \sqrt{64 + 36}}{2}\)[/tex]
This step is correct, as we calculated the discriminant correctly (64 + 36 = 100).
- Step 2: [tex]\(x = \frac{-8 \pm \sqrt{100}}{2}\)[/tex]
This step is also correct, as [tex]\(\sqrt{100} = 10\)[/tex].
- Step 3: [tex]\(x = \{-18, 2\}\)[/tex]
Here is where Soren made an error. The correct computation using the formula should result in the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
Therefore, the error was made in Step 3. Soren did not properly solve the final expressions resulting from [tex]\(\frac{-8 \pm 10}{2}\)[/tex]. The correct results should have been [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex].
