solve the quadratic equation by completing the square.
4x^2-12x-32=0
First, divide the entire equation by 4 to simplify:
x^2 - 3x - 8 = 0
Rearrange equation into suitable form:
x^2 - 3x = 8
In the equation x^2 - 3x, you'll need to add a constant to complete the square. The required constant is given by (b/2)^2. Here, b is the coefficient of x, which is -3. Hence, the constant that you'll be adding and subtracting is (-3/2)^2 = 2.25
Add the constant to both sides of the equation:
x^2 - 3x + 2.25 = 8 + 2.25
x^2 - 3x + 2.25 = 10.25
Factorise the left side of the equation to get:
(x - 1.5)^2 = 10.25
Taking the square root of both sides gives:
x - 1.5 = ± √10.25
x - 1.5 = ± 3.2
Solving for x gives the roots of the quadratic equation:
x = 1.5 + 3.2 = 4.7
x = 1.5 - 3.2 = -1.7
Step 1: Move the constant term to the right side of the equation:
4x^2 - 12x = 32
Step 2: Divide the entire equation by the coefficient of the x^2 term (4):
x^2 - 3x = 8
Step 3: Take half of the coefficient of the x term (-3/2) and square it:
(-3/2)^2 = 9/4
Step 4: Add the result obtained in Step 3 to both sides of the equation:
x^2 - 3x + 9/4 = 8 + 9/4
x^2 - 3x + 9/4 = 32/4 + 9/4
x^2 - 3x + 9/4 = 41/4
Step 5: Rewrite the left side of the equation as a perfect square trinomial:
(x - 3/2)^2 = 41/4
Step 6: Take the square root of both sides of the equation:
√[(x - 3/2)^2] = ±√(41/4)
(x - 3/2) = ±√(41)/2
Step 7: Solve for x:
x - 3/2 = ±√(41)/2
x = 3/2 ± √(41)/2
So, the solutions to the quadratic equation 4x^2 - 12x - 32 = 0 after completing the square are:
x = (3 + √(41))/2 and x = (3 - √(41))/2
To solve the quadratic equation 4x^2 - 12x - 32 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the right side of the equation.
4x^2 - 12x = 32
Step 2: Divide through by the coefficient of x^2 to make the coefficient 1.
x^2 - 3x = 8
Step 3: Take half of the coefficient of x (which is -3), square it, and add it to both sides of the equation.
x^2 - 3x + (-3/2)^2 = 8 + (-3/2)^2
x^2 - 3x + 9/4 = 8 + 9/4
Step 4: Simplify the equation on the right side.
x^2 - 3x + 9/4 = 32/4 + 9/4
x^2 - 3x + 9/4 = 41/4
Step 5: Rewrite the left side of the equation as a perfect square trinomial by factoring the square of a binomial.
(x - 3/2)^2 = 41/4
Step 6: Take the square root of both sides of the equation, remembering to consider both the positive and negative square roots.
x - 3/2 = ±√(41/4)
Step 7: Solve for x in both cases.
x = 3/2 ± √(41/4)
So, the solution to the quadratic equation 4x^2 - 12x - 32 = 0 by completing the square is:
x = 3/2 + √(41/4) or x = 3/2 - √(41/4)