Solve for all values of xx by factoring.

Question

x, squared, plus, x, minus, 61, equals, minus, 5
x

Answer 1

To solve this equation by factoring, we need to rewrite it in the form of a quadratic equation.

x^2 + x - 61 = -5

Now, let's move all the terms to one side of the equation:

x^2 + x - 61 + 5 = 0

x^2 + x - 56 = 0

To factor the quadratic expression, we need to find two numbers whose sum is equal to the coefficient of x (which is 1) and whose product is equal to the constant term (which is -56). Let's try different pairs of numbers that satisfy these conditions:

1 * (-56) = -56 (not a valid pair since the sum is not equal to 1)
2 * (-28) = -56 (not a valid pair since the sum is not equal to 1)
4 * (-14) = -56 (valid pair since the sum is equal to 1)

Now, let's rewrite the equation using these numbers:

x^2 + 4x - 14x - 56 = 0

Factor x out of the first two terms and -14 out of the last two terms:

x(x + 4) - 14(x + 4) = 0

Use the distributive property to simplify:

(x - 14)(x + 4) = 0

Now, set each factor equal to zero and solve for x:

x - 14 = 0 or x + 4 = 0

x = 14 or x = -4

Therefore, the values of x are 14 and -4.