Complete the square to identify the two values that solve the following quadratic equation: x^2 - 16x - 13x = 4.
x = __
To complete the square for the quadratic equation x^2 - 16x - 13x = 4, we need to rearrange the terms to have the constant term on the right side:
x^2 - 16x - 13x - 4 = 0
Next, we factor out the common factor from the first three terms:
x(x - 16) - 13(x - 1) = 0
Now, we can see that we have a perfect square trinomial in the first two terms and a constant term in the last term:
(x - 16)^2 - 13(x - 1) = 0
To simplify further, we distribute -13 and remove parentheses:
(x - 16)^2 - 13x + 13 = 0
Now, we can see that we have completed the square.
The two values that solve the quadratic equation are the values of x that make the equation equal to zero. Therefore,
(x - 16)^2 - 13x + 13 = 0
We can set this equation equal to zero and solve for x:
(x - 16)^2 - 13x + 13 = 0
Now, we can solve this equation by factoring, completing the square, or using the quadratic formula. Factoring this equation may be difficult, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -13, and c = 13.
x = (-(-13) ± √((-13)^2 - 4(1)(13))) / (2(1))
x = (13 ± √(169 - 52)) / 2
x = (13 ± √(117)) / 2
x = (13 ± √(9 * 13)) / 2
x = (13 ± 3√13) / 2
Therefore, the two values that solve the quadratic equation x^2 - 16x - 13x = 4 are:
x = (13 + 3√13) / 2
and
x = (13 - 3√13) / 2