The root of an equation is 5/4 and 3/2 fine the equation

Question

ScanSolve AI Tutor · Accepted Answer

Let's find the quadratic equation from its roots.

Identify the given roots.
The roots are $r_1 = \frac{5}{4}$ and $r_2 = \frac{3}{2}$ .
Calculate the sum of the roots.
$\text{Sum} = r_1 + r_2 = \frac{5}{4} + \frac{3}{2}$
To add these fractions, find a common denominator, which is 4.
$\text{Sum} = \frac{5}{4} + \frac{3 \times 2}{2 \times 2} = \frac{5}{4} + \frac{6}{4} = \frac{5+6}{4} = \frac{11}{4}$
Calculate the product of the roots.
$\text{Product} = r_1 \times r_2 = \frac{5}{4} \times \frac{3}{2}$ $\text{Product} = \frac{5 \times 3}{4 \times 2} = \frac{15}{8}$
Form the quadratic equation using the formula $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ .
Substitute the sum and product into the formula:
$x^2 - \left(\frac{11}{4}\right)x + \left(\frac{15}{8}\right) = 0$
Clear the denominators to obtain an equation with integer coefficients.
Multiply the entire equation by the least common multiple (LCM) of the denominators (4 and 8), which is 8.
$8 \left(x^2 - \frac{11}{4}x + \frac{15}{8}\right) = 8 \times 0$ $8x^2 - \left(8 \times \frac{11}{4}\right)x + \left(8 \times \frac{15}{8}\right) = 0$ $8x^2 - (2 \times 11)x + 15 = 0$ $8x^2 - 22x + 15 = 0$

Answer