Answer :

Sure! Let's solve the quadratic equation [tex]\(x^2 - 2x - 35 = 0\)[/tex] to find the values of [tex]\(x\)[/tex].

### Step-by-Step Solution:

1. Identify the coefficients.
In the equation, [tex]\(x^2 - 2x - 35 = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex] (the coefficient of [tex]\(x^2\)[/tex]),
- [tex]\(b = -2\)[/tex] (the coefficient of [tex]\(x\)[/tex]),
- [tex]\(c = -35\)[/tex] (the constant term).

2. Use the quadratic formula.
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

3. Calculate the discriminant.
The discriminant ([tex]\(\Delta\)[/tex]) is the part under the square root in the quadratic formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values, we get:
[tex]\[
\Delta = (-2)^2 - 4 \times 1 \times (-35) = 4 + 140 = 144
\][/tex]

4. Calculate the roots.
Now substitute the discriminant and the coefficients back into the quadratic formula to get the roots:
[tex]\[
x_1 = \frac{-(-2) + \sqrt{144}}{2 \times 1} = \frac{2 + 12}{2} = \frac{14}{2} = 7
\][/tex]
[tex]\[
x_2 = \frac{-(-2) - \sqrt{144}}{2 \times 1} = \frac{2 - 12}{2} = \frac{-10}{2} = -5
\][/tex]

Therefore, the solutions to the equation [tex]\(x^2 - 2x - 35 = 0\)[/tex] are [tex]\(x = 7\)[/tex] and [tex]\(x = -5\)[/tex].

Based on the given options, the correct answer is:

J. -5 and 7

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Rewritten by : Barada