Complete the square to re-write the quadratic function in vertex form:
y, equals, minus, 8, x, squared, minus, 96, x, minus, 289
y=−8x2 −96x−289
To complete the square, we want to rearrange the quadratic function to have the form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
We start by factoring out the common factor of -8 from the x^2 and x terms:
y = -8(x^2 + 12x) - 289
Next, we need to determine what number to add and subtract inside the parentheses to complete the square. The number we need to add and subtract is found by taking half of the coefficient of x and squaring it. In this case, we take half of 12 (which is 6) and square it (which gives us 36).
y = -8(x^2 + 12x + 36 - 36) - 289
Now we can rewrite the expression as:
y = -8[(x + 6)^2 - 36] - 289
Simplifying inside the brackets:
y = -8(x + 6)^2 + 288 - 289
Combine like terms:
y = -8(x + 6)^2 - 1
Therefore, the quadratic function rewritten in vertex form is:
y = -8(x + 6)^2 - 1
To complete the square and rewrite the quadratic function in vertex form, follow these steps:
Step 1: Factor out the coefficient of x^2 from the terms involving x.
y = -8(x^2 + 12x) - 289
Step 2: Take half of the coefficient of x and square it, then add it inside the parentheses.
y = -8(x^2 + 12x + 36) - 289 + 8(36)
Step 3: Simplify the equation inside the parentheses and combine like terms.
y = -8(x + 6)^2 - 289 + 288
Step 4: Further simplify the equation.
y = -8(x + 6)^2 - 1
Therefore, the quadratic function in vertex form is:
y = -8(x + 6)^2 - 1.
To rewrite the given quadratic function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by: y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Let's begin by looking at the terms involving x. We have: y = -8x^2 - 96x - 289.
To complete the square, we need to focus on the coefficient of x (in this case, -96). We divide the coefficient of x by 2, then square the result. In this case, -96 ÷ 2 = -48, and (-48)^2 = 2,304.
Next, we add and subtract 2,304 inside the parentheses (after -96x) to keep the equation balanced:
y = -8(x^2 + 96x + 2,304) - 289 - 2,304
Now, we have a perfect square trinomial inside the parentheses: x^2 + 96x + 2,304 can be rewritten as (x + 48)^2.
Plugging this back into the equation: y = -8(x + 48)^2 - 2593
So, the quadratic function can be written in vertex form as: y = -8(x + 48)^2 - 2593.