Convert equation from quadratic form to vertex form, y= a(x-h)^2 +k, by completing the square.
y= -.25x^2+5.5x-5.25
-.25x^2+5.5x-5.25
= -1/4 (x^2-22x) - 21/4
= -1/4 (x^2-22x+121) - 21/4 + 121/4
= -1/4 (x-11)^2 + 25
To convert the equation from quadratic form to vertex form by completing the square, follow these step-by-step instructions:
Step 1: Group the quadratic terms
y = (-0.25x^2 + 5.5x) - 5.25
Step 2: Factor out the coefficient of the x^2 term from the first two terms
y = -0.25(x^2 - 22x) - 5.25
Step 3: Take half of the coefficient of the x term and square it
Half of the coefficient of the x term = 22/2 = 11
Square of 11 = 121
Step 4: Add the square obtained in step 3 inside the parentheses and subtract the same value multiplied by the coefficient of the x^2 term outside the parentheses
y = -0.25(x^2 - 22x + 121 - 121) - 5.25
Step 5: Rewrite the equation inside the parentheses as a perfect square trinomial and factor it
y = -0.25((x - 11)^2 - 121) - 5.25
Step 6: Simplify the equation inside the parentheses
y = -0.25(x - 11)^2 + 30.25 - 5.25
Step 7: Combine the constants outside the parentheses
y = -0.25(x - 11)^2 + 25
The equation y = -0.25(x - 11)^2 + 25 is the vertex form of the given quadratic equation. The vertex of the parabola is (11, 25) and the coefficient "a" is -0.25.
To convert the given quadratic equation, y = -.25x^2 + 5.5x - 5.25, from standard form (quadratic form) to vertex form, we need to complete the square.
Step 1: Start with the equation in standard form: y = ax^2 + bx + c. In this case, a = -.25, b = 5.5, and c = -5.25.
Step 2: Divide the entire equation by the coefficient of x^2 (a) to make it 1. In this case, divide by -.25:
y = (-.25x^2 + 5.5x - 5.25) / -.25
This simplifies to:
y = x^2 - 22x + 21
Step 3: Move the constant term (c) to the right side of the equation:
y + 5.25 = x^2 - 22x + 21 + 5.25
This further simplifies to:
y + 5.25 = x^2 - 22x + 26.25
Step 4: Complete the square by adding the square of half the coefficient of x (b/2)^2 to both sides of the equation. In this case, the coefficient of x is -22, so the squared term is (-22/2)^2 = 11^2 = 121. Add 121 to both sides:
y + 5.25 + 121 = x^2 - 22x + 26.25 + 121
This simplifies to:
y + 126.25 = x^2 - 22x + 147.25
Step 5: Rewrite the quadratic expression as a perfect square binomial:
y + 126.25 = (x - 11)^2
Step 6: Move the constant term (126.25) to the right side of the equation:
y = (x - 11)^2 - 126.25
This is the equation in vertex form: y = a(x - h)^2 + k, where the vertex is at the point (h, k). In this case, the vertex is (11, -126.25), and a = 1.
Therefore, the equation in vertex form is:
y = (x - 11)^2 - 126.25