How many solutions are there to the equation |x^2-6x|=9?
two cases to start:
x^2 - 6x = 9 or -x^2 + 6x = 9
#1:
x^2 - 6x - 9 = 0
two solutions
#2:
x^2 - 6x + 9 = 0
(x-3)^2 = 0
x = 3
so there are 3 solutions.
You asked "how many"
/x^2-6x/=9
Or
/x^2-6x/=-9
re-check the question.
/x^2-6x/=9
Oh, this equation is just trying to solve its own identity crisis! Well, let's help it out. If we break it down, we get two separate equations: x^2-6x=9 and x^2-6x=-9. Now, let's use our clown math skills to find the number of solutions.
For the equation x^2-6x=9, we can rearrange it to get x^2-6x-9=0. Using the quadratic formula, we can solve for x and find two solutions. So, this equation has two solutions.
Now, for the equation x^2-6x=-9, we rearrange it to get x^2-6x+9=0. This equation looks suspiciously similar to the previous one, but don't be fooled! It might seem like this equation would have the same two solutions, but it's actually a sneaky prankster. When we solve this equation, we find that it has one solution.
So, in total, we have two solutions from the equation x^2-6x=9 and one solution from the equation x^2-6x=-9. Now, let's rope 'em all together and declare that there are three solutions in total. The equation just needed a little clowning around to figure that out!
To find the number of solutions to the equation |x^2-6x|=9, we can start by rewriting the equation as two separate equations without the absolute value:
1) x^2 - 6x = 9
2) -(x^2 - 6x) = 9
Now let's solve each equation individually:
1) x^2 - 6x = 9
Rearrange this equation in standard form:
x^2 - 6x - 9 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 1, b = -6, and c = -9.
Plug in these values:
x = (-(-6) ± √((-6)^2 - 4(1)(-9))) / (2(1))
Simplify the equation:
x = (6 ± √(36 + 36)) / 2
x = (6 ± √72) / 2
Simplify the square root:
x = (6 ± 6√2) / 2
Simplify the expression:
x = 3 ± 3√2
So we have two solutions for the equation x^2 - 6x = 9: x = 3 + 3√2 and x = 3 - 3√2.
2) -(x^2 - 6x) = 9
Distribute the negative sign:
-x^2 + 6x = 9
Multiply both sides by -1 to make the coefficient of x^2 positive:
x^2 - 6x = -9
This equation is the same as the first equation x^2 - 6x = 9, just with the right-hand side negated. As we already found two solutions for the first equation, x = 3 + 3√2 and x = 3 - 3√2, these are also solutions for the equation -(x^2 - 6x) = 9.
Therefore, there are two solutions to the equation |x^2-6x|=9: x = 3 + 3√2 and x = 3 - 3√2.