Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth.
X^2-6=x
To use the quadratic formula, we first need to get the equation in the form ax^2 + bx + c = 0. So we'll start by moving the 6 over to the other side:
x^2 - x - 6 = 0
Now we can identify our values:
a = 1
b = -1
c = -6
Plugging these into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-(-1) ± √((-1)^2 - 4(1)(-6))) / 2(1)
x = (1 ± √(25)) / 2
x = (1 ± 5) / 2
x = 3 or x = -2
So our solutions are x = 3 and x = -2.
Which you can check with
x^2 - x - 6 = 0
(x- 3)(x+2) = 0
x = 3 and x = -2
That's correct! We can check our solutions by plugging them back into the original equation and seeing if we get 0:
For x = 3:
x^2 - x - 6 = 0
3^2 - 3 - 6 = 0
9 - 3 - 6 = 0
0 = 0
So x = 3 is indeed a solution.
For x = -2:
x^2 - x - 6 = 0
(-2)^2 - (-2) - 6 = 0
4 + 2 - 6 = 0
0 = 0
So x = -2 is also a solution.
To solve the equation, we need to rearrange it in the standard quadratic form, which is: ax^2 + bx + c = 0.
Given equation: x^2 - 6 = x
Rearranging the terms, we get: x^2 - x - 6 = 0
Now, we can identify a, b, and c values:
a = 1
b = -1
c = -6
Using the quadratic formula, which states that the solutions for ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
we can substitute the values into the formula:
x = (-(-1) ± √((-1)^2 - 4(1)(-6))) / (2(1))
Simplifying further:
x = (1 ± √(1 + 24)) / 2
x = (1 ± √25) / 2
x = (1 ± 5) / 2
This gives us two solutions:
x1 = (1 + 5) / 2 = 6 / 2 = 3
x2 = (1 - 5) / 2 = -4 / 2 = -2
Hence, the solutions to the equation x^2 - 6 = x are x = 3 and x = -2.