Solve \sqrt{10-x} + x=8. Enter your solutions in a comma-separated list.
I used the quadratic formula and ended up with-127+-(-215)/2
Did you actually try using your values to see whether they satisfied the equation?
√(10-x) + x = 8
√(10-x) = 8-x
10-x = x^2-16x+64
x^2-15x+54 = 0
(x-9)(x-6) = 0
x = 6 or 9
should have showed your work, so we could see what went wrong.
However, since both sides were squared, all answers must be checked in the original equation
√(10-x) + x = 8
if x = 6
LS = √4 + 6 = 8 = RS
if x = 9
LS = √1 + 9 = 10 ≠ RS
so the only solution is x = 6
I did check the other answer.
Sure enough, √1 + 9 = 10
and I thought, "That's ok."
Good catch. I had expected an extraneous root, and was surprised that there "wasn't" one.
To solve the equation √(10-x) + x = 8, we can follow these steps:
Step 1: Isolate the square root term. Subtract x from both sides:
√(10-x) = 8 - x
Step 2: Square both sides to eliminate the square root:
(√(10-x))^2 = (8 - x)^2
10 - x = (8 - x)^2
Step 3: Expand the square on the right side:
10 - x = 64 - 16x + x^2
Step 4: Rearrange the equation in standard form:
x^2 - 15x + 54 = 0
Step 5: Now, we can solve the quadratic equation. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing our quadratic equation to standard quadratic form (ax^2 + bx + c = 0), we have:
a = 1, b = -15, and c = 54.
Now, substitute these values into the quadratic formula:
x = (-(-15) ± √((-15)^2 - 4(1)(54))) / (2(1))
x = (15 ± √(225 - 216)) / 2
x = (15 ± √9) / 2
x = (15 ± 3) / 2
Therefore, the solutions to the equation √(10-x) + x = 8 are:
x = (15 + 3) / 2 = 9
x = (15 - 3) / 2 = 6
The solutions can be expressed as a comma-separated list: 9, 6.