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