Complete the square to solve the following quadratic equation:
2? - 6x - 20 = 20. (1 point)
Ox = 52. 7 = -46
●¢=10.2=-4
O¢=-10.2=4
O* = 3 + 210, 3 - 2v10
To solve the quadratic equation by completing the square, we want to rearrange the equation in the form (x - a)^2 = b.
First, let's move the constant term to the other side of the equation by subtracting 20 from both sides:
2x^2 - 6x - 40 = 0
Next, divide the entire equation by the coefficient of x^2 to make the coefficient 1:
x^2 - 3x - 20 = 0
Now, to complete the square, we take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 - 3x + (-3/2)^2 - (-3/2)^2 - 20 = 0
(x - 3/2)^2 - 9/4 - 20 = 0
(x - 3/2)^2 - 89/4 = 0
Add 89/4 to both sides of the equation:
(x - 3/2)^2 = 89/4
Now, square root both sides to solve for x:
x - 3/2 = ±√(89/4)
x - 3/2 = ±(√89/2)
Add 3/2 to both sides:
x = 3/2 ± (√89/2)
Therefore, the solutions to the quadratic equation 2x^2 - 6x - 20 = 20 are x = 3/2 + (√89/2) and x = 3/2 - (√89/2).
To complete the square and solve the quadratic equation 2x^2 - 6x - 20 = 20, we can follow these steps:
Step 1: Move the constant term to the right side:
2x^2 - 6x - 20 - 20 = 0
2x^2 - 6x - 40 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the coefficient 1:
x^2 - 3x - 20 = 0
Step 3: Take half of the coefficient of x (-3), square it, and add it to both sides of the equation:
x^2 - 3x + (-3/2)^2 - 20 + (-3/2)^2 = (-3/2)^2
This becomes:
x^2 - 3x + 9/4 - 80/4 + 9/4 = 9/4
Simplifying further:
x^2 - 3x + 18/4 = 9/4
Step 4: Simplify the equation on the left side:
x^2 - 3x + 9/4 = 9/4
Step 5: Rewrite the left side as a perfect square trinomial:
(x - 3/2)^2 = 9/4
Step 6: Take the square root of both sides (plus/minus):
x - 3/2 = ±√(9/4)
Step 7: Simplify the right side:
x - 3/2 = ±(3/2)
Step 8: Solve for x:
Case 1: x - 3/2 = 3/2
x = 3/2 + 3/2
x = 6/2
x = 3
Case 2: x - 3/2 = -3/2
x = -3/2 + 3/2
x = 0/2
x = 0
Therefore, the solutions to the quadratic equation 2x^2 - 6x - 20 = 20 are x = 3 and x = 0.
To complete the square and solve the quadratic equation 2x^2 - 6x - 20 = 20, follow these steps:
1. Move the constant term to the other side of the equation:
2x^2 - 6x - 20 - 20 = 0
Simplify:
2x^2 - 6x - 40 = 0
2. Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1:
(2x^2 - 6x - 40) / 2 = 0
Simplify:
x^2 - 3x - 20 = 0
3. Now, focus on the quadratic term and the linear term. Take half of the coefficient of the linear term (-3 in this case) and square it:
(-3/2)^2 = 9/4
4. Add the square obtained in the previous step to both sides of the equation:
x^2 - 3x - 20 + 9/4 = 9/4
Simplify:
x^2 - 3x + 9/4 - 80/4 = 9/4
x^2 - 3x + 1/4 = 9/4
5. Rewrite the left side of the equation as a perfect square trinomial:
(x - 3/2)^2 = 9/4
6. Take the square root of both sides of the equation:
√((x - 3/2)^2) = ±√(9/4)
Simplify:
x - 3/2 = ±(3/2)
7. Solve for x:
x - 3/2 = 3/2 or x - 3/2 = -3/2
For the first equation:
x = 3/2 + 3/2
Simplify:
x = 6/2
x = 3
For the second equation:
x = 3/2 - 3/2
Simplify:
x = 0/2
x = 0
So, the solutions to the equation 2x^2 - 6x - 20 = 20 after completing the square are x = 3 and x = 0.