Solve the quadratic equation 7x^2 - 14x = -7 Show your work
To solve the quadratic equation 7x^2 - 14x = -7, we rearrange the equation to have zero on one side:
7x^2 - 14x + 7 = 0
Next, we can factor out a common factor of 7 from each term:
7(x^2 - 2x + 1) = 0
Now, we can try to factor the quadratic expression inside the parentheses. We are looking for two numbers that multiply to give 1 (the constant term) and add to give -2 (the coefficient of the middle term). The numbers -1 and -1 meet these criteria, so we can factor the quadratic as:
7(x - 1)(x - 1) = 0
Simplifying further, we have:
7(x - 1)^2 = 0
At this point, we have two options:
1) Apply the zero product property: If the product of two factors is equal to zero, then at least one of the factors must be equal to zero. So either x - 1 = 0 (giving x = 1) or 7 = 0. However, 7 is not equal to zero, so the only solution is x = 1.
2) Use the square root property: Let (x - 1)^2 = 0. Taking the square root of both sides, we have x - 1 = 0, which gives x = 1.
Therefore, the solution to the quadratic equation 7x^2 - 14x = -7 is x = 1.