Math Problem Statement

Solve this

If a > 0 and x = 3 in the equation shown, what is the value of a? √(3a² - 3) - x = 0

Solution

The equation in the image is:

3a23x=0\sqrt{3a^2 - 3} - x = 0

We are given x=3x = 3 and a>0a > 0. Let's solve for aa.

Step 1: Substitute x=3x = 3 into the equation

3a233=0\sqrt{3a^2 - 3} - 3 = 0

Step 2: Add 3 to both sides

3a23=3\sqrt{3a^2 - 3} = 3

Step 3: Square both sides to eliminate the square root

3a23=93a^2 - 3 = 9

Step 4: Add 3 to both sides

3a2=123a^2 = 12

Step 5: Divide both sides by 3

a2=4a^2 = 4

Step 6: Take the square root of both sides

a=2a = 2

Since a>0a > 0, the solution is a=2a = 2.

Thus, the value of aa is 2.

Would you like further details or explanations on any step?

Here are 5 related questions to expand on this concept:

  1. How would the solution change if aa could be negative?
  2. What happens if you substitute x=5x = 5 instead of x=3x = 3?
  3. How would the problem change if the equation was 3a23+x=0\sqrt{3a^2 - 3} + x = 0?
  4. Can you solve a similar equation where the square root is on both sides, like 3a23=9\sqrt{3a^2 - 3} = \sqrt{9}?
  5. How do you solve for aa if the equation involved a cubic term, like 3a33=3\sqrt{3a^3 - 3} = 3?

Tip: Always verify your solution by substituting the value of the variable back into the original equation to check for accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Square Roots
Solving Equations

Formulas

√(3a² - 3) - x = 0
a² = 4

Theorems

Properties of Square Roots

Suitable Grade Level

Grades 9-11