Complete the square to solve the following quadratic equation: x2−6x−20=20 .(1 point) Responses x=10 , x=−4 x equals 10 , x equals negative 4 x=3+210−−√, 3−210−−√ x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root x=−10 , x=4 x equals negative 10 , x equals 4 x=52 , x=−46
To complete the square, we need to move the constant term to the right side of the equation:
x^2 - 6x - 20 = 20
Subtracting 20 from both sides:
x^2 - 6x - 40 = 0
To complete the square, we take half of the coefficient of x (-6), square it, and add it to both sides:
x^2 - 6x + (-6/2)^2 = 40 + (-6/2)^2
x^2 - 6x + 9 = 40 + 9
Simplifying:
x^2 - 6x + 9 = 49
Now, we can rewrite the left side as a perfect square:
(x - 3)^2 = 49
Taking the square root of both sides:
x - 3 = ±√49
x - 3 = ±7
Solving for x:
x = 3 + 7 = 10
x = 3 - 7 = -4
Therefore, the solutions are x = 10 and x = -4.
To complete the square and solve the quadratic equation x^2 - 6x - 20 = 20, follow these steps:
1. Move the constant term (in this case, 20) to the other side of the equation:
x^2 - 6x - 20 - 20 = 0
Simplify to get: x^2 - 6x - 40 = 0
2. To complete the square, focus on the x terms. Take half the coefficient of x (-6), square it, and add it to both sides of the equation:
x^2 - 6x + (-6/2)^2 = 40 + (-6/2)^2
Simplify: x^2 - 6x + 9 = 40 + 9
3. Simplify the right side of the equation:
x^2 - 6x + 9 = 49
4. Rewrite the left side of the equation as a perfect square trinomial by factoring:
(x - 3)^2 = 49
5. Take the square root of both sides:
√((x - 3)^2) = ±√49
6. Remove the square root and simplify:
x - 3 = ±7
7. Solve for x:
x = 3 + 7 or x = 3 - 7
This gives us two solutions: x = 10 or x = -4.
So, the correct answer is x = 10 and x = -4.
To complete the square to solve the quadratic equation x^2 - 6x - 20 = 20, you can follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 - 6x - 20 - 20 = 0
x^2 - 6x - 40 = 0
Step 2: Divide the coefficient of x (in this case, -6) by 2 and square the result:
(-6/2)^2 = 9
Step 3: Add the result from step 2 to both sides of the equation:
x^2 - 6x + 9 - 40 = 9
x^2 - 6x + 9 - 40 + 9 = 9 + 9
x^2 - 6x + 9 = 18
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
(x - 3)^2 = 18
Step 5: Take the square root of both sides of the equation to solve for x:
√((x - 3)^2) = ±√18
x - 3 = ±√18
Step 6: Add 3 to both sides of the equation to isolate x:
x - 3 + 3 = ±√18 + 3
x = 3 ± √18
Therefore, the solutions to the quadratic equation x^2 - 6x - 20 = 20, after completing the square, are:
x = 3 + √18 (approximately 7.24)
and
x = 3 - √18 (approximately -1.24)