Complete the square to identify the vertex of y=x2−16x+14 .(1 point) Responses (8,30) left parenthesis 8 comma 30 right parenthesis (8,78) left parenthesis 8 comma 78 right parenthesis (8,−50) left parenthesis 8 comma negative 50 right parenthesis (8,14) left parenthesis 8 comma 14 right parenthesis
To complete the square for the given quadratic equation y=x^2 - 16x + 14, we need to follow these steps:
Step 1: Take half of the coefficient of x, and square it.
Half of the coefficient of x is -16/2 = -8.
Square of -8 is (-8)^2 = 64.
Step 2: Add the value obtained in Step 1 to both sides of the equation.
y + 64 = x^2 - 16x + 64 + 14
Simplifying, we get:
y + 64 = x^2 - 16x + 78
Step 3: Rewrite the right side of the equation as a perfect square trinomial.
The right side of the equation can be written as:
y + 64 = (x - 8)^2 + 78
Step 4: Subtract 64 from both sides of the equation.
y + 64 - 64 = (x - 8)^2 + 78 - 64
Simplifying, we get:
y = (x - 8)^2 + 14
Comparing the equation in Step 4 to the vertex form of a parabola, y = (x - h)^2 + k,
we can see that the vertex of the given quadratic equation is at the point (h, k) = (8, 14).
Therefore, the correct answer is:
(8, 14)
To complete the square to identify the vertex of the quadratic equation y=x^2-16x+14, you can follow these steps:
Step 1: Group the terms containing x together:
y = (x^2 - 16x) + 14
Step 2: Take half of the coefficient of x (-16) and square it:
Half of -16 is -8, and its square is 64.
Step 3: Add and subtract the value obtained in step 2 inside the parentheses:
y = (x^2 - 16x + 64 - 64) + 14
Step 4: Rewrite the expression by grouping the perfect square trinomial with the constant term outside the parentheses:
y = ((x - 8)^2 - 64) + 14
Step 5: Simplify the equation:
y = (x - 8)^2 - 64 + 14
y = (x - 8)^2 - 50
So, the completed square form of the equation is y = (x - 8)^2 - 50.
From this form, we can identify the vertex of the parabola as the point (8, -50).
Therefore, the correct answer is (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) and divide it by 2, then square the result. In this case, (-16/2)^2 = 64.
Step 2: Add the result from Step 1 to both sides of the equation. y = x^2 - 16x + 14 + 64. This step completes the square.
Step 3: Simplify the equation. y = x^2 - 16x + 78.
Step 4: Rearrange the equation to factor it. y = (x - 8)^2 - 64 + 78.
Step 5: Simplify further. y = (x - 8)^2 + 14.
Based on the simplified form of the equation, we can identify the vertex as (8, 14). Therefore, the correct answer to the question is (8, 14) in the format of a coordinated pair.