17. Solve 3x² - V3x =8

Answer 1

To solve the equation 3x² - √3x = 8, we can follow these steps:

1. Move all terms to one side of the equation: 3x² - √3x - 8 = 0.

2. Since the coefficient of the x² term is 3, we need to multiply 3 with the constant term (-8). We get -24.

3. We need to find two numbers that multiply to -24 and add up to -√3. The numbers -6 and 4 satisfy these conditions since -6 * 4 = -24 and -6 + 4 = -2 = -√3.

4. Rewrite the equation by splitting the middle term using the two numbers found in the previous step: 3x² - 6√3x + 4√3x - 8 = 0.

5. Factor by grouping: (3x² - 6√3x) + (4√3x - 8) = 0.

6. Factor out common terms from each group: 3x(x - 2√3) + 4(x - 2) = 0.

7. Factor out common terms from the entire equation: (x - 2)(3x + 4√3) = 0.

8. Set each factor to zero: x - 2 = 0 or 3x + 4√3 = 0.

9. Solve each equation separately:
x - 2 = 0 -> x = 2.
3x + 4√3 = 0 -> 3x = -4√3 -> x = -4√3/3.

Therefore, the solutions are x = 2 and x = -4√3/3.