Complete the square to solve the following quadratic equation: x^2−6x−20=20
To complete the square, we need to manipulate the equation so that the left side is a perfect square trinomial.
First, let's move the constant term to the right side:
x^2 - 6x - 20 + 20 = 20 + 20
This simplifies to:
x^2 - 6x = 40
To complete the square, we need to take half the coefficient of x (-6), square it, and add it to both sides of the equation.
Half of -6 is -3, and squaring it gives us 9. So, we add 9 to both sides:
x^2 - 6x + 9 = 40 + 9
This simplifies to:
(x - 3)^2 = 49
Now, we can take the square root of both sides to solve for x:
√((x - 3)^2) = √(49)
Simplifying this yields:
x - 3 = ± 7
Now, we solve for x by adding 3 to both sides:
x - 3 + 3 = 7 + 3
This gives us one solution:
x = 10
And subtracting 3 from both sides gives us:
x - 3 + 3 = -7 + 3
This gives us another solution:
x = -4
Therefore, the solutions to the quadratic equation x^2 - 6x - 20 = 20 are x = 10 and x = -4.
To complete the square, first move the constant term (the number that is not multiplied by x) to the right side of the equation:
x^2 - 6x = 20 + 20
x^2 - 6x = 40
Now, take half of the coefficient of x (-6) and square it:
(-6/2)^2 = 9
Add this value to both sides of the equation:
x^2 - 6x + 9 = 40 + 9
x^2 - 6x + 9 = 49
Now, rewrite the left side of the equation as a perfect square trinomial:
(x - 3)^2 = 49
Take the square root of both sides of the equation:
x - 3 = ±√49
x - 3 = ±7
Now, solve for x by splitting the equation into two separate equations:
x - 3 = 7 or x - 3 = -7
For the first equation:
x = 7 + 3
x = 10
And for the second equation:
x = -7 + 3
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. This involves manipulating the equation to express it in a perfect square trinomial form, and then solving for x.
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: Take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 - 6x + (-6/2)^2 = 40 + (-6/2)^2
x^2 - 6x + 9 = 40 + 9
x^2 - 6x + 9 = 49
Step 3: Write the left side of the equation as a perfect square trinomial:
(x - 3)^2 = 49
Step 4: Take the square root of both sides to eliminate the square:
√((x - 3)^2) = ±√49
(x - 3) = ±7
Step 5: Solve for x by considering both the positive and negative square roots:
If (x - 3) = 7, then x = 7 + 3 = 10
If (x - 3) = -7, then x = -7 + 3 = -4
Therefore, the solutions to the quadratic equation x^2 - 6x - 20 = 20 are x = 10 and x = -4.