Complete the square to solve the following quadratic equation:
X^2 – 22x + 1 = 15
To complete the square, we need to rearrange the equation so that the constant term is on the right side:
X^2 - 22x = 15 - 1
X^2 - 22x = 14
Next, take half of the coefficient of x (-22) and square it. In this case, ((-22)/2)^2 = 11^2 = 121.
Add 121 to both sides of the equation:
X^2 - 22x + 121 = 14 + 121
X^2 - 22x + 121 = 135
Now, the left side of the equation can be factored as a perfect square:
(X - 11)^2 = 135
To solve for x, take the square root of both sides:
X - 11 = ±√135
X - 11 = ±√(9 * 15)
X - 11 = ±3√15
Finally, solve for x:
X = 11 ± 3√15
To complete the square and solve the quadratic equation X^2 - 22x + 1 = 15, follow these steps:
Step 1: Move the constant term to the right side of the equation:
X^2 - 22x + 1 - 15 = 0
X^2 - 22x - 14 = 0
Step 2: Divide the coefficient of the x term by 2, and then square the result. Add this squared value to both sides of the equation:
X^2 - 22x + (-22/2)^2 = 14 + (-22/2)^2
X^2 - 22x + 121 = 14 + 121
X^2 - 22x + 121 = 135
Step 3: Rewrite the left side of the equation as a perfect square trinomial:
(X - 11)^2 = 135
Step 4: Take the square root of both sides of the equation:
√((X - 11)^2) = √(135)
X - 11 = ±√135
Step 5: Solve for X by adding or subtracting 11 from both sides of the equation:
X = 11 ±√135
Therefore, the solutions to the quadratic equation X^2 - 22x + 1 = 15 are:
X = 11 + √135
X = 11 - √135
To complete the square and solve the quadratic equation, follow these steps:
Step 1: Move the constant term (in this case, 15) to the right side of the equation:
X^2 – 22x + 1 - 15 = 0
Simplifying the equation further:
X^2 – 22x - 14 = 0
Step 2: Divide the coefficient of the x-term (in this case, -22) by 2, and square the result:
(-22 / 2)^2 = (-11)^2 = 121
Step 3: Add the square from step 2 to both sides of the equation:
X^2 – 22x - 14 + 121 = 121
Simplifying the equation further:
X^2 – 22x + 107 = 121
Step 4: Rearrange the equation to create a perfect square trinomial on the left side:
(X^2 – 22x + 121) + 107 = 121
Simplifying the equation further:
(X - 11)^2 + 107 = 121
Step 5: Subtract the constant term added in step 4 from both sides of the equation:
(X - 11)^2 = 121 - 107
Simplifying the equation further:
(X - 11)^2 = 14
Step 6: Take the square root of both sides of the equation:
√[(X - 11)^2] = √14
Simplifying the equation further:
X - 11 = ±√14
Step 7: Solve for x by adding 11 to both sides of the equation:
X = 11 ± √14
So, the solutions to the quadratic equation X^2 – 22x + 1 = 15 are X = 11 + √14 and X = 11 - √14.