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Answer :
To solve the quadratic equation [tex]\(x^2 + 12x + 35 = 0\)[/tex] using the Quadratic Formula, we use the formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation, the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 12\)[/tex]
- [tex]\(c = 35\)[/tex]
Let's calculate each step:
1. Calculate the Discriminant:
The discriminant is given by the part under the square root in the quadratic formula: [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
b^2 = 12^2 = 144
\][/tex]
[tex]\[
4ac = 4 \times 1 \times 35 = 140
\][/tex]
[tex]\[
\text{Discriminant} = b^2 - 4ac = 144 - 140 = 4
\][/tex]
2. Find the Square Root of the Discriminant:
[tex]\[
\sqrt{4} = 2
\][/tex]
3. Calculate the Two Possible Solutions:
Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute the values we have:
[tex]\[
x = \frac{-12 \pm 2}{2 \times 1}
\][/tex]
First Solution:
[tex]\[
x_1 = \frac{-12 + 2}{2} = \frac{-10}{2} = -5
\][/tex]
Second Solution:
[tex]\[
x_2 = \frac{-12 - 2}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 + 12x + 35 = 0\)[/tex] are [tex]\(x = -5\)[/tex] and [tex]\(x = -7\)[/tex]. This matches the choice:
[tex]\(x = -5, -7\)[/tex]
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation, the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 12\)[/tex]
- [tex]\(c = 35\)[/tex]
Let's calculate each step:
1. Calculate the Discriminant:
The discriminant is given by the part under the square root in the quadratic formula: [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
b^2 = 12^2 = 144
\][/tex]
[tex]\[
4ac = 4 \times 1 \times 35 = 140
\][/tex]
[tex]\[
\text{Discriminant} = b^2 - 4ac = 144 - 140 = 4
\][/tex]
2. Find the Square Root of the Discriminant:
[tex]\[
\sqrt{4} = 2
\][/tex]
3. Calculate the Two Possible Solutions:
Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute the values we have:
[tex]\[
x = \frac{-12 \pm 2}{2 \times 1}
\][/tex]
First Solution:
[tex]\[
x_1 = \frac{-12 + 2}{2} = \frac{-10}{2} = -5
\][/tex]
Second Solution:
[tex]\[
x_2 = \frac{-12 - 2}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 + 12x + 35 = 0\)[/tex] are [tex]\(x = -5\)[/tex] and [tex]\(x = -7\)[/tex]. This matches the choice:
[tex]\(x = -5, -7\)[/tex]
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