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complete the square to solve the following quadratic equation x^2-22x+1-15

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
1 year ago

To complete the square, let's focus on the quadratic term, which is x^2 - 22x.

First, we need to take half of the coefficient of the x-term (-22) and square it: (-22/2)^2 = 121.

Next, we add this value to both sides of the equation:
x^2 - 22x + 121 = 1 - 15 + 121

Simplifying further on the right side:
x^2 - 22x + 121 = 107

Now, we can factor the left side as a perfect square:
(x - 11)^2 = 107

To solve for x, we take the square root of both sides:
√(x - 11)^2 = √107

This leads to two possible solutions:
x - 11 = √107 or x - 11 = -√107

Solving for x in each equation:
x = 11 + √107 or x = 11 - √107

Therefore, the solutions to the quadratic equation x^2 - 22x + 1 - 15 = 0 are x = 11 + √107 and x = 11 - √107.

User Icon for Explain Bot Explain Bot answered
1 year ago

To complete the square, you need to rewrite the quadratic equation in the form of (x - h)^2 = k, where h and k are constants. This will allow you to easily find the value(s) of x that satisfy the equation.

Let's start with the given quadratic equation:

x^2 - 22x + 1 - 15 = 0

First, combine the constant terms:

x^2 - 22x - 14 = 0

To complete the square, we will focus on the first two terms, x^2 - 22x. To find the term that needs to be added to both sides of the equation to create a perfect square trinomial, take half of the coefficient of x (-22 in this case), square it, and add it to both sides of the equation.

Step 1: Divide the coefficient of x by 2 (-22/2 = -11).
Step 2: Square the result (-11)^2 = 121.
Step 3: Add the result to both sides of the equation.

x^2 - 22x + 121 - 14 = 121

The left side can now be rewritten as a perfect square trinomial:

(x - 11)^2 - 14 = 121

Move the constant term to the right side of the equation:

(x - 11)^2 = 121 + 14

Simplify:

(x - 11)^2 = 135

Now, take the square root of both sides to eliminate the exponent:

√[(x - 11)^2] = ±√135

Solve for x:

x - 11 = ±√135

Add 11 to both sides:

x = 11 ± √135

Therefore, the solutions for the quadratic equation x^2 - 22x + 1 - 15 = 0, after completing the square, are:

x = 11 + √135 and x = 11 - √135

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

To complete the square and solve the quadratic equation x^2 - 22x + 1 - 15 = 0, follow these steps:

Step 1: Move the constant term to the other side of the equation:
x^2 - 22x + 1 - 15 = 0
x^2 - 22x - 14 = 0

Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation. In this case, the coefficient of x is -22, so:
x^2 - 22x + (-22/2)^2 = 14 + (-22/2)^2
x^2 - 22x + 121 = 14 + 121

Step 3: Simplify both sides of the equation:
x^2 - 22x + 121 = 135

Step 4: Rewrite the left side of the equation as a perfect square:
(x - 11)^2 = 135

Step 5: Take the square root of both sides of the equation:
√(x - 11)^2 = ±√135

Step 6: Simplify the square root of 135:
(x - 11) = ±√(9 * 15)
(x - 11) = ±3√15

Step 7: Solve for x by adding 11 to both sides:
x = 11 ± 3√15

Therefore, the solutions to the quadratic equation x^2 - 22x + 1 - 15 = 0, after completing the square, are:
x = 11 + 3√15 and x = 11 - 3√15.