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: Move the constant term to the other side of the equation:
2x^2 + 5x = 8
Step 2: Divide the entire equation by the coefficient of x^2 to make the coefficient of x^2 equal to 1:
x^2 + (5/2)x = 4
Step 3: Take half of the coefficient of x (in this case, 5/2), square it, and add it to both sides of the equation:
x^2 + (5/2)x + (5/4)^2 = 4 + (5/4)^2
Simplifying:
x^2 + (5/2)x + (25/16) = 4 + (25/16)
Step 4: Simplify the right side of the equation:
x^2 + (5/2)x + (25/16) = 64/16 + 25/16
x^2 + (5/2)x + (25/16) = 89/16
Step 5: Factor the left side of the equation.
(x + 5/4)^2 = 89/16
Step 6: Take the square root of both sides of the equation to isolate x:
x + 5/4 = ± √(89/16)
Step 7: Solve for x:
x = -5/4 ± √(89/16)
Simplifying, we get two solutions:
x = (-5 ± √89) / 4
To solve the quadratic equation 2x^2 + 5x - 8 = 0 by completing the square, follow these steps:
1. Move the constant term to the other side of the equation:
2x^2 + 5x = 8
2. Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1:
(2x^2 + 5x)/2 = 8/2
x^2 + (5/2)x = 4
3. Take half of the coefficient of x and square it. Add this result to both sides of the equation:
x^2 + (5/2)x + (5/4)^2 = 4 + (5/4)^2
x^2 + (5/2)x + 25/16 = 64/16 + 25/16
x^2 + (5/2)x + 25/16 = 89/16
4. Factor the left side of the equation:
(x + 5/4)^2 = 89/16
5. Take the square root of both sides:
sqrt((x + 5/4)^2) = sqrt(89/16)
x + 5/4 = ± sqrt(89)/4
6. Solve for x by subtracting 5/4 from both sides:
x = -5/4 ± sqrt(89)/4
Therefore, the quadratic equation 2x^2 + 5x - 8 = 0 is solved by completing the square, giving the solutions x = (-5 ± sqrt(89))/4.
x^2 + (5/2)x = 4
x^2 + (5/2)x + 25/16 = 4 + 25/16
(x + 5/4)^2 = 89/16
x + 5/4 = ± √89/4
x = (-5 ± √89)/4