Complete the square to identify the vertex of y = x^2 – 16x + 14.
To complete the square and identify the vertex of the equation y = x^2 - 16x + 14, we first need to take half of the coefficient of x and square it.
Half of -16 is -8, and (-8)^2 is 64.
To complete the square, we add 64 to both sides of the equation:
y + 64 = x^2 - 16x + 14 + 64
Simplifying the right side:
y + 64 = x^2 - 16x + 78
Next, we can rewrite the right side as a perfect square trinomial by factoring the quadratic expression:
y + 64 = (x - 8)^2 + 78 - 64
Simplifying further:
y + 64 = (x - 8)^2 + 14
Finally, we subtract 64 from both sides to isolate y:
y + 64 - 64 = (x - 8)^2 + 14 - 64
y = (x - 8)^2 - 50
The completed square form of the equation is y = (x - 8)^2 - 50.
From this form, we can identify that the vertex of the parabola is at (8, -50).
To complete the square and find the vertex of the quadratic equation y = x^2 – 16x + 14, follow these steps:
Step 1: Take the coefficient of x (which is -16), divide it by 2, and then square it:
(-16/2)^2 = (-8)^2 = 64
Step 2: Add this result to both sides of the equation:
y + 64 = x^2 - 16x + 14 + 64
Simplify:
y + 64 = x^2 - 16x + 78
Step 3: Rewrite the right side of the equation as a perfect square trinomial:
y + 64 = (x - 8)^2
Step 4: To isolate y, subtract 64 from both sides of the equation:
y = (x - 8)^2 - 64
The equation is now in vertex form, y = (x - h)^2 + k, where the vertex is represented by the point (h, k). Comparing our equation to the vertex form, we can conclude that the vertex of the quadratic equation y = x^2 – 16x + 14 is (8, -64).
Completing the square is a method used to rewrite a quadratic equation in the form '(x - h)^2 + k', where (h, k) represents the coordinates of the vertex of the parabola. Here's how to complete the square for the given equation:
Step 1: Start with the equation in the form y = ax^2 + bx + c. In this case, we have y = x^2 - 16x + 14.
Step 2: Take the coefficient of the x term (b), divide it by 2, and square the result. In this case, -16 divided by 2 equals -8, and (-8)^2 equals 64.
Step 3: Add the result from step 2 to both sides of the equation. We add 64 to both sides of the equation y = x^2 - 16x + 14 + 64.
So far, we have y + 64 = x^2 - 16x + 78.
Step 4: Rewrite the first three terms on the right side of the equation as a perfect square trinomial. To achieve this, we need to find the square root of the coefficient of the x term, divide it by 2, square the result (which gives 4), and add it as a binomial factor inside the parentheses.
So, we have y + 64 = (x^2 - 16x + 64) + 14.
Step 5: Simplify the equation by combining like terms. The trinomial (x^2 - 16x + 64) can be factored as (x - 8)^2.
Therefore, the equation becomes y + 64 = (x - 8)^2 + 14.
Step 6: Finally, subtract 64 from both sides to isolate the squared term. This gives us y = (x - 8)^2 + 14 - 64, which simplifies to y = (x - 8)^2 - 50.
We can now identify the vertex of the parabola from the equation: the vertex is located at the point (h, k) = (8, -50). So, the vertex of y = x^2 - 16x + 14 is (8, -50).