Factoring... x^2 + 4x - 6 = 0?

Question

1.) Factor: x^2 + 4x - 6 = 0

2.) Find the center and the radius of the circle whose equation is given by: x^2 + y^2 + 10x -56 = 0
(Hint: use completeing the square)

Answer 1

Completing the square

x² + 4x - 6 = 0

x² + 4x - 6 + 6 = 0 + 6

x² + 4x = 6

x² + 4x + _____ = 6 + ______

x² + 4x + 4 = 6 + 4

(x + 2)(x + 2) = 10

(x + 2)² = 10

(√x + 2)² = ± √10

x + 2 = ± √10

x + 2 - 2 = - 2 ± 3.16227766

x = - 2 ± 3.16227766

- - - - - - - - - -

Solving for +

x = - 2 + 3.16227766

x = 1.6227766

- - - - - - - - -

Solving for -

x = 5.16227766

- - - - - - - - - - -s-

Answer 2

for 1), find two numbers that multiply to -6 and add to +4
(or the product is 6 and the difference is 4.)
1*6, 2*3; neither has a difference of 4. So we try to complete the square
x^2 + 4x = 6
x^2 + 4x + 4 = 10
(x + 2)^2 = 10
x + 2 = ±sqrt(10)
x = -2 ±sqrt(10)
so its factors are (x + 2 + 10^(1/2))(x + 2 - 10^(1/2))
check: x^2 + 2x - xrt10 + 2x + 4 - 2*rt10 + xrt10 + 2*rt10 - 10
= x^2 +2x + 2x + 4 - 10 = x^2 + 4x - 6.
Thus this ugly factoring is the correct answer to the question you posted. (I suspect the question is wrong, since math books usually have sanitized, nice, and pretty answers)

x^2 + y^2 + 10x - 56 = 0
There is no y^1 term, so we leave y^2 as a completed square. Our x term, however involves an x^2 + 10x. The rest of the square is (10/2)^2 = 25. thus we have
(x + 5)^2 + y^2 - 56 - 25 = 0
(x+5)^2 + y^2 = 81 which is of the form
(x - h)^2 + (y - k)^2 = r^2 --the equation of a circle centered at (h, k) with radius r.
So the center is (-5, 0) and radius is sqrt(81) = 9.

Answer 3

1. The equation is not factorable with integer roots.

2.
We should rewrite the equation x^2 + y^2 + 10x -56 = 0
as: x^2 +10x - 56 + y^2 = 0

Now add 25 to both sides of the equation.

x^2 + 10x + 25 - 56 +y^2 = 0

Then we can rewrite it as:
(x+5)^2 - 56 + y^2 = 25

Move the 56 to the other side of the equation.
(x+5)^2 + y^2 = 81

Remember that the general form for a circle is
(x-h)^2 + (y-k)^2 = r^2

Where r is the radius and the center of the circle is (h,k)

So from the equation we can gather that:
The radius is 9 and the center is (-5,0)