sandy wrote:
If then \(x^2 - x \sqrt{2} + 3x\sqrt{3}=\sqrt{54}\) then \(x =\)
Indicate all possible values.
A. \(-\sqrt{2}\)
B. \(-3\sqrt{2}\)
C. \(-3\sqrt{3}\)
D. \(\sqrt{2}\)
One option is to plug in each answer choice and see if it satisfies the given equation.
It's time-consuming, but works.
Here's another approach:
GIVEN: x² - x√2 + 3x√3 = √54
Simplify √54
√54 = √[(9)(6)] = (√9)(√6) = 3√6
So, we can write: x² - x√2 + 3x√3 = 3√6
Subtract 3√6 from both sides to get: x² - x√2 + 3x√3 - 3√6 = 0
From here, we'll use an advanced technique called
FACTORING IN PARTS.
Factor the greatest common factor from the first two terms to get:
x(
x - √2) + 3x√3 - 3√6 = 0
Factor the last two terms to get:
x(
x - √2) +
3√3(
x - √2) = 0
COMBINE the terms to get: (
x + 3√3)(
x - √2) = 0
From here, we can see that the solutions are: x = -3√3 and x = √2
Answer: C, D
Cheers,
Brent