Question

Then solve the equation: `x^{2}-2x-35=0` for x.

Please show all of your work writing below or in the math response box.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the equation `x^{2}-2x-35=0`, we can factorize the quadratic expression or use the quadratic formula.

1. Factoring:
We need to find two numbers whose sum is -2 and whose product is -35. The numbers -7 and 5 satisfy this condition, as -7 + 5 = -2 and -7 * 5 = -35. Therefore, we can factorize the equation as:
(x - 7)(x + 5) = 0

Setting each factor equal to 0:
x - 7 = 0 or x + 5 = 0

Solving each equation for x:
x = 7 or x = -5

2. Quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac))/(2a)

For our equation x^{2}-2x-35=0, the coefficients are:
a = 1, b = -2, c = -35

Substituting these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4*1*(-35)))/(2*1)
x = (2 ± √(4 + 140))/2
x = (2 ± √144)/2
x = (2 ± 12)/2

Simplifying:
x = (2 + 12)/2 or x = (2 - 12)/2
x = 14/2 or x = -10/2
x = 7 or x = -5

Therefore, the solutions to the equation `x^{2}-2x-35=0` are x = 7 and x = -5.