Complete the square to solve the following quadratic equation: x^2 - 6x - 20 = 20.
x^2 - 6x - 20 = 20 can be rewritten as:
x^2 - 6x - 40 = 0.
To complete the square, take half of the coefficient of x and square it. Half of -6 is -3, and (-3)^2 = 9.
Add 9 to both sides of the equation to maintain balance:
x^2 - 6x + 9 = 40 + 9
x^2 - 6x + 9 = 49.
The left side of the equation can be factored as a perfect square:
(x - 3)^2 = 49.
Take the square root of both sides to solve for x:
x - 3 = ±√49
x - 3 = ±7.
Now, solve for x by adding 3 to both sides of the equation:
x = 3 + 7 or x = 3 - 7.
The two solutions to the equation x^2 - 6x - 20 = 20 are x = 10 or x = -4.
To complete the square, we need to take the quadratic equation in the form of Ax^2 + Bx + C = 0.
Given equation: x^2 - 6x - 20 = 20.
Step 1: Move the constant term to the right side of the equation:
x^2 - 6x - 20 - 20 = 0
x^2 - 6x - 40 = 0
Step 2: We want the coefficient of x^2 to be 1. Since it is already 1 in this equation, we don't need to make any changes.
Step 3: Take half of the coefficient of x and square it. Add that result to both sides of the equation.
Coefficient of x = -6/2 = -3
(-3)^2 = 9
Adding 9 to both sides of the equation:
x^2 - 6x - 40 + 9 = 0 + 9
x^2 - 6x - 31 = 9
Step 4: Rewrite the equation as a perfect square trinomial.
We can rewrite the left side of the equation as a perfect square trinomial by factoring it. We take half of the coefficient of x, square it, and then add it and subtract it from the equation.
x^2 - 6x + (-3)^2 - (-3)^2 - 31 = 9
(x^2 - 6x + 9) - 9 - 31 = 9
(x - 3)^2 - 40 = 9
Step 5: Simplify the equation.
(x - 3)^2 - 40 = 9
Step 6: Move the constant term to the right side of the equation.
(x - 3)^2 - 40 + 40 = 9 + 40
(x - 3)^2 = 49
Step 7: Take the square root of both sides.
√((x - 3)^2) = √49
x - 3 = ±7
Step 8: Solve for x.
We solve for x by adding 3 to both sides of the equation separately for the positive and negative roots.
x - 3 = 7 or x - 3 = -7
x = 7 + 3 or x = -7 + 3
x = 10 or x = -4
Therefore, the solutions to the quadratic equation x^2 - 6x - 20 = 20 are x = 10 and x = -4.
To solve the quadratic equation x^2 - 6x - 20 = 20, we can complete the square.
Step 1: Move the constant term to the right side of the equation:
x^2 - 6x - 20 - 20 = 0
Simplifying the equation gives:
x^2 - 6x - 40 = 0
Step 2: To complete the square, focus on the coefficient of the x-term, which is -6. Divide it by 2 and square the result:
(-6 / 2)^2 = 9
Step 3: Add the result from Step 2 (9) to both sides of the equation:
x^2 - 6x + 9 - 40 = 9
Simplifying the equation gives:
x^2 - 6x + 9 - 40 = 9
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
(x - 3)^2 = 9
Step 5: Take the square root of both sides of the equation:
√(x - 3)^2 = ±√9
Simplifying the equation gives:
x - 3 = ±3
Step 6: Solve for x by adding 3 to both sides of the equation:
x = 3 ± 3
Therefore, the solutions to the quadratic equation x^2 - 6x - 20 = 20 are:
x = 3 + 3 = 6
x = 3 - 3 = 0