Final answer:
To put the equation x^2 + 14x - 9 = 6 in vertex form, you need to complete the square. First, divide the equation by the coefficient of x^2. Then, add and subtract the square of half the coefficient of x. Group the first three terms and simplify the equation to isolate the perfect square trinomial. The vertex form of the equation is y = (x + 7)^2 - 58.
Step-by-step explanation:
This expression is a quadratic equation of the form at² + bt + c = 0, where the constants are a = 1, b = 14, and c = -9. To put the equation in vertex form, we need to complete the square. The vertex form of a quadratic equation is given by y = a(x-h)² + k, where (h, k) represents the vertex of the parabola.
Let's first divide the equation by the coefficient of x², which is 1. We get x² + 14x - 9 = 6. To complete the square, we need to add and subtract the square of half the coefficient of x.
Half of 14 is 7, and the square of 7 is 49. So, we can add and subtract 49 inside the parentheses.
x² + 14x + 49 - 49 - 9 = 6
In the vertex form, we want to express the equation as a perfect square trinomial. So, we group the first three terms and simplify the equation:
(x² + 14x + 49) - 49 - 9 = 6
(x + 7)² - 58 = 6
Finally, we can add 58 to both sides of the equation to isolate the perfect square trinomial:
(x + 7)² = 64
The equation is now in vertex form, and we have found the vertex to be (-7, 64). The vertex form of the equation is y = (x + 7)² - 58.