Solve the equation by completing the square. If necessary, round to the nearest hundredth.
x² – 18x = 19
A. 1; 19
B. –1; 19
C. 3; 6
D. –3; 1
First, we need to move the constant over to the right side:
x² – 18x + _____ = 19 + _____
To figure out what goes in the blank, we take half of the coefficient of x (-18) and square it:
(-18/2)² = 81
So we add 81 to both sides:
x² – 18x + 81 = 100
Now we can write the left side as a square:
(x – 9)² = 100
Taking the square root of both sides, we get:
x – 9 = ±10
Adding 9 to both sides:
x = 9 ± 10
So the solutions are:
x = 19 or x = –1
Therefore, the answer is B. –1; 19.
To solve the equation by completing the square, we follow these steps:
1. Move the constant term to the other side of the equation:
x² – 18x = 19 becomes x² – 18x - 19 = 0
2. Take half of the coefficient of the x-term and square it:
(-18/2)² = 81
3. Add this value to both sides of the equation:
x² – 18x + 81 = 19 + 81
x² – 18x + 81 = 100
4. Rewrite the left side of the equation as a perfect square binomial:
(x - 9)² = 100
5. Take the square root of both sides of the equation:
√[(x - 9)²] = √100
(x - 9) = ±√100
x - 9 = ±10
6. Solve for x by adding 9 to both sides of the equation:
x - 9 + 9 = ±10 + 9
x = 9 ± 10
So, the solutions to the equation are:
x = 9 + 10 = 19
x = 9 - 10 = -1
Rounded to the nearest hundredth, the solutions are:
A. 1; 19