Solve each equation by completing the square.
-36x=8x^2+28
-8x^2-3x=-5
5x^2-x-6=0
6x^2-24x-30=0
7x^2+8x-12=0
-10x^2 -9x+1=0
5x^2-x=4
-11x^2-8x=-7x-10
-3 1/3x^2+ 10x -27 1/2= -231/3
Show me how you did one of them, and I will evaluate your solution.
BTW, with the coefficients of your questions, the method of completing the square would be my last choice of the 3 main methods of solving quadratic equations.
To solve each of the given equations by completing the square, the following steps can be followed:
1. Rearrange the equation so that the terms with variables are on one side and the constant terms are on the other side. Make sure the quadratic term has a coefficient of 1. If not, divide the entire equation by the coefficient of the quadratic term.
2. Move the constant term to the other side of the equation.
3. Write the equation in the form: (x + h)^2 = k, where h and k are constants.
4. Take half of the coefficient of the x term and square it to get h.
5. Add h^2 to both sides of the equation.
6. Rewrite the equation as a perfect square trinomial on the left side.
7. Take the square root of both sides of the equation. The positive and negative roots will be considered separately.
8. Solve for x by subtracting h from the square root expression on both sides.
Now let's solve each of the provided equations:
Equation 1: -36x = 8x^2 + 28
Step 1: Rearrange the equation: 8x^2 + 36x + 28 = 0
Step 2: Move the constant term to the other side: 8x^2 + 36x = -28
Step 3: Write the equation in the form: 8(x^2 + 4.5x) = -28
Step 4: Find h: h = (4.5/2)^2 = 2.25^2 = 5.0625
Step 5: Add h^2 to both sides of the equation: 8(x^2 + 4.5x + 5.0625) = -28 + 8(5.0625)
Step 6: Rewrite the equation as a perfect square trinomial: 8(x + 2.25)^2 = 16
Step 7: Take the square root of both sides: x + 2.25 = ±√(16/8) = ±2
Step 8: Solve for x: x = -2.25 ± 2
So the solutions to the equation are x = -4.25 and x = 0.75.
Repeat the above steps for the remaining equations to find their solutions.