Question

Find the roots of the quadratic equation \( \sqrt{3}x^2 - 11x + 8\sqrt{3} = 0 \).

Hint:

When dealing with quadratic equations involving square roots, the splitting the middle term method can be tricky. If you're struggling to find the right factors, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a reliable alternative that will always work.

Answer 1

Step 1: Understanding the Concept:
We need to solve a quadratic equation. This can be done by factoring (splitting the middle term) or by using the quadratic formula.
Step 2: Key Formula or Approach:
We will use the method of splitting the middle term. We need to find two numbers that sum to the coefficient of the middle term (-11) and multiply to the product of the first and last coefficients (\( \sqrt{3} \times 8\sqrt{3} \)).
Step 3: Detailed Explanation:
The given equation is \( \sqrt{3}x^2 - 11x + 8\sqrt{3} = 0 \).
Product of the first and last coefficients = \( (\sqrt{3}) \times (8\sqrt{3}) = 8 \times 3 = 24 \).
Sum of the coefficients for the middle term = -11.
We need to find two numbers whose product is 24 and whose sum is -11. These numbers are -3 and -8.
Now, split the middle term:
\[ \sqrt{3}x^2 - 3x - 8x + 8\sqrt{3} = 0 \] Factor by grouping. Note that \( 3 = \sqrt{3} \times \sqrt{3} \).
\[ \sqrt{3}x(x - \sqrt{3}) - 8(x - \sqrt{3}) = 0 \] Take out the common factor \( (x - \sqrt{3}) \):
\[ (x - \sqrt{3})(\sqrt{3}x - 8) = 0 \] This gives two possible solutions:
Either \( x - \sqrt{3} = 0 \), which means \( x = \sqrt{3} \).
Or \( \sqrt{3}x - 8 = 0 \), which means \( \sqrt{3}x = 8 \), so \( x = \frac{8}{\sqrt{3}} \).
Rationalizing the denominator for the second root: \( x = \frac{8}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \).
Step 4: Final Answer:
The roots of the quadratic equation are \( \sqrt{3} \) and \( \frac{8\sqrt{3}}{3} \).