Select the correct vertex form of the following equation. show all your work
y=x2+6x−9
answer : y=(x−3)^2−4
To convert the given equation into vertex form, we need to complete the square.
First, let's group the terms:
y = x^2 + 6x - 9
Next, we want to add and subtract a constant inside the parentheses to complete the square. To determine the constant, take half of the coefficient of x and square it. In this case, half of 6 is 3, and 3 squared is 9.
y = x^2 + 6x + 9 - 9 - 9
Now, we can rewrite the equation as:
y = (x^2 + 6x + 9) - 9 - 9
Now, let's factor the square trinomial in the parentheses:
y = (x + 3)^2 - 18
Simplifying further, we have:
y = (x + 3)^2 - 18
Therefore, the correct vertex form of the equation is y = (x - 3)^2 - 4.
To find the vertex form of the given quadratic equation, we need to complete the square. Here are the steps to do that:
Step 1: Start with the given quadratic equation:
y = x^2 + 6x - 9
Step 2: Take the coefficient of x (which is 6), divide it by 2, and square the result. Add that value to both sides of the equation:
y + 9 = x^2 + 6x + (6/2)^2
This becomes:
y + 9 = x^2 + 6x + 9
Step 3: Rewrite the first three terms as a perfect square trinomial. Factor it and simplify:
y + 9 = (x + 3)^2
Step 4: Subtract 9 from both sides to isolate y:
y = (x + 3)^2 - 9
Step 5: Simplify the equation further:
y = (x + 3)^2 - 9
The vertex form of the equation is y = (x + 3)^2 - 9. Therefore, the correct answer is y = (x - 3)^2 - 4.
To find the vertex form of the equation y = x^2 + 6x - 9, we need to complete the square. Here is the step-by-step procedure:
Step 1: Separate the constant term and the coefficient of x:
y = (x^2 + 6x) - 9
Step 2: Move the constant term to the right side:
y + 9 = x^2 + 6x
Step 3: Take half of the coefficient of x (which is 6) and square it (36/4 = 9):
y + 9 + 9 = x^2 + 6x + 9
Step 4: Rewrite the right side as a perfect square:
y + 18 = (x^2 + 6x + 9)
Step 5: Convert the right side expression into vertex form by factoring the perfect square:
y + 18 = (x + 3)^2
Step 6: Move the constant term to the left side by subtracting 18 from both sides:
y = (x + 3)^2 - 18
The correct vertex form of the equation y = x^2 + 6x - 9 is y = (x - 3)^2 - 18.