Solve the quadratic equation by completing the square. Verify your answer graphically. (Enter your answers as a comma-separated list.)
16x^2 − 24x − 23 = 0
16x^2 − 24x − 23 = 0
x^2 - (24/16)x = 23/16
x^2 - (3/2)x + 9/16 = 23/16 + 9/16
(x - 3/4)^2 = 2
x - 3/4 = ± √2
x = 3/4 ± √2
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
16x^2 - 24x = 23
Step 2: Divide the entire equation by the coefficient of the x^2 term to make the coefficient equal to 1:
x^2 - (24/16)x = 23/16
Step 3: Take half of the coefficient of the x term (24/16) and square it:
(24/16)/2 = (3/2)^2 = 9/4
Step 4: Add the result from step 3 to both sides of the equation:
x^2 - (24/16)x + 9/4 = 23/16 + 9/4
Simplifying the right side:
23/16 + 9/4 = (23/16) + (36/16) = 59/16
Step 5: Factor the left side of the equation:
(x - 3/4)^2 = 59/16
Step 6: Take the square root of both sides of the equation:
√[(x - 3/4)^2] = ±√(59/16)
Simplifying the left side:
x - 3/4 = ±(√59/4)
Step 7: Solve for x:
x = 3/4 ± (√59/4)
The solutions to the quadratic equation 16x^2 - 24x - 23 = 0, when completing the square, are x = 3/4 + (√59/4) and x = 3/4 - (√59/4).
To verify the solutions graphically, you can plot the quadratic equation on a graphing calculator or a plotting tool. Plot the equation 16x^2 - 24x - 23 = 0 and see where it intersects the x-axis. The x-coordinates of those points of intersection will verify the solutions obtained algebraically.