2x^2 + 5x - 8 = 0
Solve the quadratic equation by completing the square.
To solve the quadratic equation 2x^2 + 5x - 8 = 0 by completing the square, follow these steps:
Step 1: Make sure the coefficient of x^2 is 1. In this equation, the coefficient is already 2, so divide the entire equation by 2 to make it x^2 + (5/2)x - 4 = 0.
Step 2: Move the constant term to the other side of the equation. Add 4 to both sides: x^2 + (5/2)x = 4.
Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation. Half of (5/2) is 5/4, and (5/4)^2 is 25/16. Add 25/16 to both sides: x^2 + (5/2)x + 25/16 = 4 + 25/16.
Step 4: Rewrite the left side of the equation as a perfect square trinomial. The left side factors to (x + 5/4)^2.
Step 5: Simplify the right side of the equation. Find a common denominator for 4 and 16, which is 16. 4 + 25/16 becomes (64/16) + (25/16) = 89/16.
Step 6: Write the equation in its completed square form. The equation is now (x + 5/4)^2 = 89/16.
Step 7: Take the square root of both sides of the equation. Remember to consider both the positive and negative square roots.
Positive square root: x + 5/4 = sqrt(89)/4
Negative square root: x + 5/4 = -sqrt(89)/4
Step 8: Solve for x in each equation.
Positive square root: x = -5/4 + sqrt(89)/4
Negative square root: x = -5/4 - sqrt(89)/4
So the solutions to the quadratic equation 2x^2 + 5x - 8 = 0 by completing the square are:
x = -5/4 + sqrt(89)/4
x = -5/4 - sqrt(89)/4