Makeup a quadratic equation in standard form with the following properties: a not equal to 1 or 0, b not equal to 0 and c not equal to 0. Complete the square to change the relation to vertex form. *
y = 4x^2 + 8x + 5 = 4(x+1)^2 + 1
3 x^2 + 6 x - 18 = y
x^2 + 2 x - 6 = y/3 (getting a = 1)
x^2 + 2 x = y/3 + 6
[(1/2)* 2 ]^2 = 1 so add 1 to both sides
x^2 + 2 x + 1 = y/3 + 7
(x+1)^2 = (1/3)( y + 21)
looks like vertex at (-1 , -21)
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To create a quadratic equation in standard form with the given properties, we need to follow a few steps:
Step 1: Choose a non-zero value for "a" (coefficient of the x^2-term).
Step 2: Choose a non-zero value for "b" (coefficient of the x-term).
Step 3: Choose a non-zero value for "c" (constant term).
For this example, let's choose:
a = 2
b = 3
c = 4
The quadratic equation in standard form is:
2x^2 + 3x + 4 = 0
Now, to complete the square and change the equation to vertex form, we can follow these steps:
Step 1: Move the constant term (c) to the other side of the equation to isolate the squared and linear terms.
2x^2 + 3x = -4
Step 2: Divide the entire equation by the coefficient of the x^2-term (a) to make it equal to 1.
x^2 + (3/2)x = -2
Step 3: Take half of the coefficient of the x-term (b) and square it [(b/2)^2].
(3/2)^2 = 9/4
Step 4: Add the value obtained in Step 3 to both sides of the equation.
x^2 + (3/2)x + 9/4 = -2 + 9/4
Step 5: Simplify the equation.
x^2 + (3/2)x + 9/4 = -8/4 + 9/4
x^2 + (3/2)x + 9/4 = 1/4
Step 6: Factorize the square term on the left side of the equation.
(x + (3/4))^2 = 1/4
Step 7: Take the square root of both sides of the equation.
x + (3/4) = ±√(1/4)
Step 8: Solve for x.
x + (3/4) = ±1/2
Case 1: x + (3/4) = 1/2
x = 1/2 - (3/4)
x = 1/2 - 3/4
x = -1/4
Case 2: x + (3/4) = -1/2
x = -1/2 - (3/4)
x = -1/2 - 3/4
x = -5/4
Therefore, the quadratic equation in standard form is:
2x^2 + 3x + 4 = 0
And in vertex form, it is:
(x + (3/4))^2 = 1/4
Simplified: x^2 + (3/2)x + 9/4 = 1/4
The vertex of the quadratic equation is (-3/4, 1/4).