Form the quadratic whose roots are 2/sqrt(3) and sqrt(3).

Question

ScanSolve AI Tutor · Accepted Answer

To form a quadratic equation given its roots, we use the formula $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ .

Identify the given roots.
Let the roots be $\alpha = \frac{2}{\sqrt{3}}$ and $\beta = \sqrt{3}$ .
Calculate the sum of the roots.
$\alpha + \beta = \frac{2}{\sqrt{3}} + \sqrt{3}$
To add these, we find a common denominator:
$\alpha + \beta = \frac{2}{\sqrt{3}} + \frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} = \frac{2}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{2+3}{\sqrt{3}} = \frac{5}{\sqrt{3}}$
Rationalize the denominator:
$\alpha + \beta = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$
Calculate the product of the roots.
$\alpha \beta = \frac{2}{\sqrt{3}} \cdot \sqrt{3} = 2$
Substitute the sum and product into the quadratic equation formula.
The general form of a quadratic equation is $x^2 - (\alpha + \beta)x + \alpha\beta = 0$ .
$x^2 - \left(\frac{5\sqrt{3}}{3}\right)x + 2 = 0$
Clear the denominator by multiplying the entire equation by 3.
$3 \left(x^2 - \frac{5\sqrt{3}}{3}x + 2\right) = 3 \cdot 0$ $3x^2 - 5\sqrt{3}x + 6 = 0$

Answer