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Math  /  Algebra

Question2−−3x2−3=3x−92--3 \sqrt{x^{2}-3}=3 x-9

Studdy Solution

STEP 1

1. The equation involves a square root and linear terms.
2. The domain of the solution must ensure that the expression inside the square root is non-negative.
3. Solving the equation will involve isolating the square root and then squaring both sides to eliminate it.

STEP 2

1. Isolate the square root term.
2. Square both sides to eliminate the square root.
3. Solve the resulting equation.
4. Check for extraneous solutions due to squaring.

STEP 3

Start by isolating the square root term. We have:
2−(−3x2−3)=3x−9 2 - (-3\sqrt{x^2 - 3}) = 3x - 9
Simplify the left side by removing the double negative:
2+3x2−3=3x−9 2 + 3\sqrt{x^2 - 3} = 3x - 9
Subtract 2 from both sides to further isolate the square root:
3x2−3=3x−11 3\sqrt{x^2 - 3} = 3x - 11

STEP 4

Divide both sides by 3 to completely isolate the square root:
x2−3=x−113 \sqrt{x^2 - 3} = x - \frac{11}{3}

STEP 5

Square both sides to eliminate the square root:
(x2−3)2=(x−113)2 (\sqrt{x^2 - 3})^2 = \left(x - \frac{11}{3}\right)^2
This simplifies to:
x2−3=(x−113)2 x^2 - 3 = \left(x - \frac{11}{3}\right)^2

STEP 6

Expand the right side:
x2−3=x2−223x+1219 x^2 - 3 = x^2 - \frac{22}{3}x + \frac{121}{9}

STEP 7

Subtract x2 x^2 from both sides:
−3=−223x+1219 -3 = -\frac{22}{3}x + \frac{121}{9}

STEP 8

Multiply through by 9 to clear the fractions:
−27=−66x+121 -27 = -66x + 121

STEP 9

Add 66x to both sides and then subtract 121 from both sides:
66x=121−27 66x = 121 - 27
66x=94 66x = 94

STEP 10

Divide both sides by 66 to solve for x x :
x=9466 x = \frac{94}{66}
Simplify the fraction:
x=4733 x = \frac{47}{33}

STEP 11

Check the solution in the original equation to ensure it is valid and not extraneous. Substitute x=4733 x = \frac{47}{33} back into the original equation and verify both sides are equal.
The solution is:
4733 \boxed{\frac{47}{33}}

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