Solve the quadratic equation by completing the square.
x^2-18x-6=0
To solve using completing the square, let's first group the x terms:
x^2 - 18x = 6
To complete the square, we take half the coefficient of x (which is -18) and square it, which is 81. We add this to both sides:
x^2 - 18x + 81 = 6 + 81
Simplify:
(x-9)^2 = 87
Take the square root of both sides:
x - 9 = ± sqrt(87)
Finally, solve for x to give:
x = 9 ± sqrt(87)
Therefore, the roots of the equation are:
x = 9 + sqrt(87) and x = 9 - sqrt(87).
Step 1: Move the constant term to the other side of the equation:
x^2 - 18x = 6
Step 2: Take half of the coefficient of the x-term and square it. In this case, half of -18 is -9, and (-9)^2 = 81.
Add this squared value to both sides of the equation to maintain balance:
x^2 - 18x + 81 = 6 + 81
Simplified equation: x^2 - 18x + 81 = 87
Step 3: The left side of the equation can be factored as a perfect square:
(x - 9)^2 = 87
Step 4: Take the square root of both sides of the equation:
√(x - 9)^2 = ±√87
Simplified equation: x - 9 = ±√87
Step 5: Solve for x by adding 9 to both sides of the equation:
x = 9 ±√87
Therefore, the solutions to the quadratic equation x^2 - 18x - 6 = 0 are x = 9 + √87 and x = 9 - √87.
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 - 18x = 6
Step 2: Divide the coefficient of x^2 by 2 and then square it. Add this value to both sides of the equation:
x^2 - 18x + (18/2)^2 = 6 + (18/2)^2
x^2 - 18x + 81 = 6 + 81
Step 3: Simplify the equation:
x^2 - 18x + 81 = 87
Step 4: Rewrite the left side of the equation as a perfect square:
(x - 9)^2 = 87
Step 5: Take the square root of both sides of the equation:
√(x - 9)^2 = ±√87
Step 6: Solve for x:
x - 9 = ±√87
Step 7: Add 9 to both sides of the equation:
x = 9 ± √87
So the solutions to the quadratic equation x^2 - 18x - 6 = 0 are x = 9 + √87 and x = 9 - √87.