Expert: Scott A Wilson - 7/2/2011

Question
"I am having difficulty with a few problems in my college algebra class. Thank you so much for your help.

1. (x-2)^2=20

What are the exact solutions? i got -4+squaroot of 80/2 and -4-squareroot of 8.944/2
x=

What are the approximate solutions? i got 2,6
x=

2. find the x-intercept and y-intercepts for f(x). f(x)=x^2-6x+1

x-intercepts are
y-intercept is

i got x-intercepts are (0,6)
I got y-intercept 0

3. Give exact and approximate solutions to three decimal places. x^2+18x+81=1

x=

I got x=-9,-9

4. solve by completing the square. x^2+13/2x=13
x=

I got x=squarroot of 377/16, -squarroot of 377/16"

Answer
1. (x-2)² = 20; taking the square root of both sides gives x-2 =  ±√20.
Now 20 = 4*5 and √4 = ±2, ±√20 = ±2√5, so this is x-2 = ±2√5.
Adding 2 to both sides gives x = 2 ± 2√5 = 2(1 ± √5).
Since √5 is roughly 2.236, this would be -2.472 and 6.472.

2. find the x-intercept and y-intercepts for f(x). f(x) = x² - 6x + 1
To do so, we must factor the equation.  The factors of 6 are 2 and 3.
Since the 6 is negative and the one is positive, the factors are (x-3)(x+2).

Checking this gives x² - 3x + 2x – 6.  Now -3x + 2x = –x, so this is x² - x -6.

From the factored form, the x factors are 3 and -2.

From the multiplied out form, the y-intercept is the last term in the multiplied out expression.
That is because that is what is gotten when x=0, so it is at y = -6.

3. Give exact and approximate solutions to three decimal places. x² + 18x + 81 = 1
This factors to (x-9)² = 1, so x-9 = ±1.  Adding 9 to both sides gives x = 9±1.
This is the same as x = 8 or x = 10.

4. Solve by completing the square. x²+13/2x=13; is that really (x²+13)/2x=13?  It could be read as x²  +  13/2x, but the x² and 13 are close by, so I assume it is the first choice given.  Given that, multiply both side by 2x.  This gives x² + 13 = 26x.

Completing the square means 1st to put the x terms on one side and the constant on the other, so we have x² - 26x = -13.

Once this is done, take -26/3 and get 13, so add 13² to both sides; note that 13² = 169.

The equation become x² - 26x + 169 = 169 – 13.
Since the left is (x-13)² and on the right, 169 – 13 = 156, we have (x-13)² = 156.

Taking the square-root of both sides gives x-13 = ±√156 = ±√(4*39) = ±2√39.

This means we have x-13 = ±2√39.
Add 13 to both sides and get x = 13 ±2√39.
Since the square root of 39 is about 6.245, the answer is x = 13 ± 12.49.
This gives x = 0.51 or x = 25.49.