y=x^2+6x-9
select the correct vertex form
show the steps to get it
To convert the equation from standard form to vertex form, follow these steps:
1. Complete the square by adding and subtracting the square of half the coefficient of x in the equation.
y = x^2 + 6x - 9
First, take half of the coefficient of x (6/2 = 3) and square it (3^2 = 9). Add and subtract 9 within the parentheses:
y = x^2 + 6x + 9 - 9 - 9
2. Rearrange the equation by grouping the perfect square terms and the constant term separately:
y = (x^2 + 6x + 9) - 9 - 9
3. Simplify the grouped terms:
y = (x^2 + 6x + 9) - 18
4. Factor the perfect square trinomial within the parentheses:
y = (x + 3)(x + 3) - 18
5. Simplify further within the parentheses:
y = (x + 3)^2 - 18
Therefore, the equation y = x^2 + 6x - 9 is in vertex form: y = (x + 3)^2 - 18.
The vertex form of the equation is (x + 3)^2 - 18.
To express the quadratic equation y = x^2 + 6x - 9 in vertex form, follow these steps:
Step 1: Complete the square on the x terms.
To complete the square, take half of the coefficient of x (which is 6) and square it. Half of 6 is 3, and 3 squared is 9. Add 9 inside the brackets to both sides of the equation, but since we added 9, we need to subtract 9*1 at the same time.
Thus, the equation becomes:
y + 9 - 9 = x^2 + 6x + 9 - 9
Which simplifies as:
y + 9 = (x^2 + 6x + 9) - 9
Step 2: Factor the perfect square trinomial.
The trinomial x^2 + 6x + 9 can be factored as (x + 3)^2.
So, our equation becomes:
y + 9 = (x + 3)^2 - 9
Step 3: Isolate y.
To isolate y, subtract 9 from both sides of the equation:
y + 9 - 9 = (x + 3)^2 - 9 - 9
y = (x + 3)^2 - 18
Therefore, the vertex form of the quadratic equation y = x^2 + 6x - 9 is y = (x + 3)^2 - 18.
To find the vertex form of the quadratic equation y = x^2 + 6x - 9, we need to complete the square.
Step 1: Group the x^2 and x terms together
y = (x^2 + 6x) - 9
Step 2: Take half of the coefficient of the x-term (which is 6) and square it.
(6/2)^2 = 3^2 = 9
Step 3: Add and subtract the value obtained in step 2 inside the parentheses.
y = (x^2 + 6x + 9 - 9) - 9
Step 4: Rearrange the equation by grouping the perfect square trinomial.
y = (x^2 + 6x + 9) - 9 - 9
Step 5: Simplify the equation by expanding the perfect square trinomial and combining like terms.
y = (x + 3)^2 - 18
Therefore, the correct vertex form of the given quadratic equation is y = (x + 3)^2 - 18.