Solve the quadratic function y=2x^2-14x -16 by completing the square. Round roots to the nearest hundredth, if necessary. Show your work.
Look for two factors for each of the two end terms.
16 = 4,4 or 8,2 or 16,1
2 = only 2,1
By combining them, what will add to -14?
Remember that one has to be negative.
That is the beginning of your work. Carry on.
2x^2-14x -16 = 0
(x^2-7x) = 8
(x^2-7x+(7/2)^2) = 8 + (7/2)^2
(x - 7/2)^2 = 32/4 + 49/4
(x - 7/2)^2 = 81/4
x - 7/2 = ±9/2
x = 7/2 ± 9/2
x = -1,8
To solve the quadratic equation by completing the square, we need to follow these steps:
Step 1: Set the equation to zero
We start by setting the given equation equal to zero:
2x^2 - 14x - 16 = 0
Step 2: Divide the equation by the leading coefficient
Since the leading coefficient is 2, divide the entire equation by 2:
x^2 - 7x - 8 = 0
Step 3: Move the constant term to the other side of the equation
Now, move the constant term (-8) to the right side of the equation:
x^2 - 7x = 8
Step 4: Complete the square
To complete the square, take half of the coefficient of the x-term (-7/2) and square it to get (49/4):
x^2 - 7x + (49/4) = 8 + (49/4)
Step 5: Simplify the equation
Now, simplify the right side of the equation:
x^2 - 7x + (49/4) = 32/4 + 49/4
Step 6: Rewrite the left side as a perfect square trinomial
On the left side, rewrite the equation as a perfect square trinomial:
(x - (7/2))^2 = 81/4
Step 7: Take the square root of both sides
Take the square root of both sides of the equation:
√((x - (7/2))^2) = √(81/4)
Step 8: Solve for x
Solving for x, we have two cases:
Case 1: x - (7/2) = (9/2)
x = (9/2) + (7/2)
x = 16/2
x = 8
Case 2: x - (7/2) = -(9/2)
x = -(9/2) + (7/2)
x = -2/2
x = -1
Therefore, the solutions to the quadratic equation y = 2x^2 - 14x - 16 are x = 8 and x = -1.