complete the square to solve the following quadratic equation x^2-22x+1-15
x=-11+3
To complete the square for the quadratic equation x^2 - 22x + 1 - 15, first group the x terms together and isolate the constant term:
x^2 - 22x + (-15 + 1).
Combine like terms:
x^2 - 22x - 14.
To complete the square, take half of the coefficient of x (-22) and square it:
(-22/2)^2 = (-11)^2 = 121.
Add this value inside the parentheses and subtract it outside the parentheses:
x^2 - 22x + 121 - 14 - 121.
Simplify inside the parentheses:
(x - 11)^2 - 135.
Therefore, the equation x^2 - 22x + 1 - 15 can be rewritten as (x - 11)^2 - 135.
To solve for x, set the equation equal to zero:
(x - 11)^2 - 135 = 0.
Add 135 to both sides:
(x - 11)^2 = 135.
Take the square root of both sides:
√((x - 11)^2) = ±√135.
Simplify:
x - 11 = ±√(9 * 15).
x - 11 = ±√9 * √15.
x - 11 = ±3√15.
Finally, isolate x:
x = 11 ± 3√15.
Therefore, the solutions for the quadratic equation x^2 - 22x + 1 - 15 are x = 11 + 3√15 and x = 11 - 3√15.
To complete the square and solve the quadratic equation x^2 - 22x + 1 - 15 = 0, follow these steps:
1. Move the constant term (-15) to the right side of the equation:
x^2 - 22x + 1 = 15
2. Take half of the coefficient of x (-22) and square it:
(-22/2)^2 = (-11)^2 = 121
3. Add the value obtained in step 2 to both sides of the equation:
x^2 - 22x + 121 + 1 = 15 + 121
x^2 - 22x + 122 = 136
4. Rewrite the left side of the equation as a perfect square trinomial:
(x - 11)^2 = 136
5. Take the square root of both sides of the equation:
√((x - 11)^2) = √(136)
6. Simplify the square root on the right side:
x - 11 = ± √(136)
7. Add 11 to both sides of the equation:
x = 11 ± √(136)
So, the two solutions to the quadratic equation x^2 - 22x + 1 - 15 = 0 are:
x = 11 + √(136) and x = 11 - √(136)
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 - 14 = 0
Step 2: Identify the coefficient of the x term, which is -22, and divide it by 2. In this case, -22/2 = -11.
Step 3: Square the result obtained in step 2. (-11)^2 = 121.
Step 4: Add the squared result obtained in step 3 to both sides of the equation:
x^2 - 22x - 14 + 121 = 121
x^2 - 22x + 107 = 121
Step 5: Factor the perfect square trinomial on the left side of the equation, which is x^2 - 22x + 107:
(x - 11)(x - 11) = 121
Step 6: Take the square root of both sides of the equation:
√[(x - 11)(x - 11)] = ±√121
Simplifying:
x - 11 = ±11
Step 7: Solve for x by adding the constant term to both sides of the equation:
x = 11 + 11
x = 22
Step 8: Check the solution by substituting it back into the original equation:
(22)^2 - 22(22) + 1 - 15 = 0
484 - 484 + 1 - 15 = 0
0 = 0
Therefore, the solution to the quadratic equation x^2 - 22x + 1 - 15 = 0 is x = 22.