Solve by completing the square.
1. 2x^2=5x+12
2. 5x^2+10x-7=0
x^2 - (5/2)x = 6
x^2 - (5/2) x + 25/16 = 96/16 + 25/16
(x-5/4)^2 = 121/16
x-5/4 = +/- 11/4
x = (1/4)( 5 +/- 11)
To solve equations by completing the square, follow these steps:
1. Move the constant term to the other side of the equation, if necessary, so that the equation is in the form ax^2 + bx + c = 0.
Now, let's solve the given equations:
1. 2x^2 = 5x + 12
Step 1: Move the constant term to the other side to get the equation in the form ax^2 + bx + c = 0.
2x^2 - 5x - 12 = 0
Step 2: Divide the entire equation by the coefficient of x^2 if it is not 1. In this case, the coefficient is already 2, so we can skip this step.
Step 3: Complete the square by adding the square of half the coefficient of x to both sides of the equation. The coefficient of x is -5, so half of it is -5/2, and its square is 25/4.
2x^2 - 5x + 25/4 - 25/4 - 12 = 0
Step 4: Simplify the equation.
2(x^2 - (5/2)x) + 25/4 - 48/4 = 0
2(x - 5/4)^2 - 23/4 = 0
Step 5: Move the constant term to the other side.
2(x - 5/4)^2 = 23/4
Step 6: Divide both sides by the coefficient of the squared term.
(x - 5/4)^2 = 23/8
Step 7: Take the square root of both sides, remembering to consider both the positive and negative square roots.
x - 5/4 = ±√(23/8)
Step 8: Simplify the equation.
x = 5/4 ± √(23/8)
Therefore, the solution to the equation 2x^2 = 5x + 12 after completing the square is:
x = 5/4 ± √(23/8)
2. 5x^2 + 10x - 7 = 0
Step 1: Move the constant term to the other side to get the equation in the form ax^2 + bx + c = 0.
5x^2 + 10x = 7
Step 2: Divide the entire equation by the coefficient of x^2 if it is not 1. In this case, the coefficient is already 5, so we can skip this step.
Step 3: Complete the square by adding the square of half the coefficient of x to both sides of the equation. The coefficient of x is 10, so half of it is 10/2 = 5, and its square is 25.
5x^2 + 10x + 25 = 7 + 25
Step 4: Simplify the equation.
5(x^2 + 2x + 5) = 32
Step 5: Move the constant term to the other side.
5(x + 1)^2 = 32
Step 6: Divide both sides by the coefficient of the squared term.
(x + 1)^2 = 32/5
Step 7: Take the square root of both sides, remembering to consider both the positive and negative square roots.
x + 1 = ±√(32/5)
Step 8: Simplify the equation.
x = -1 ± √(32/5)
Therefore, the solution to the equation 5x^2 + 10x - 7 = 0 after completing the square is:
x = -1 ± √(32/5)