Form a quadratic equation where roots are 3/4 and 1/3, giving your answer in the form ax2 + bx + c = 0…

Question

Form a quadratic equation where roots are 3/4 and 1/3, giving your answer in the form ax2 + bx + c = 0, whereas a, b, and c are integers.

ScanSolve AI Tutor · Accepted Answer

Here's how to form the quadratic equation:

22) Form a quadratic equation where roots are $\frac{3}{4}$ and $\frac{1}{3}$ giving your answer in the form $ax^2 + bx + c = 0$ , whereas $a$ , $b$ and $c$ are integers.

Identify the given roots.
The roots are $x_1 = \frac{3}{4}$ and $x_2 = \frac{1}{3}$ .
Write the factors corresponding to these roots.
If $x = \frac{3}{4}$ , then $x - \frac{3}{4} = 0$ .
If $x = \frac{1}{3}$ , then $x - \frac{1}{3} = 0$ .
Form the quadratic equation by multiplying the factors.
$\left(x - \frac{3}{4}\right)\left(x - \frac{1}{3}\right) = 0$
Expand the product:
$x\left(x - \frac{1}{3}\right) - \frac{3}{4}\left(x - \frac{1}{3}\right) = 0$ $x^2 - \frac{1}{3}x - \frac{3}{4}x + \left(-\frac{3}{4}\right)\left(-\frac{1}{3}\right) = 0$ $x^2 - \frac{1}{3}x - \frac{3}{4}x + \frac{3}{12} = 0$ $x^2 - \frac{1}{3}x - \frac{3}{4}x + \frac{1}{4} = 0$
Combine the $x$ terms and clear the denominators to get integer coefficients.
Find a common denominator for the fractions $\frac{1}{3}$ and $\frac{3}{4}$ , which is 12.
$x^2 - \frac{4}{12}x - \frac{9}{12}x + \frac{1}{4} = 0$ $x^2 - \frac{13}{12}x + \frac{1}{4} = 0$
To eliminate the fractions, multiply the entire equation by the least common multiple of the denominators (12 and 4), which is 12.
$12\left(x^2 - \frac{13}{12}x + \frac{1}{4}\right) = 12(0)$ $12x^2 - 12\left(\frac{13}{12}\right)x + 12\left(\frac{1}{4}\right) = 0$ $12x^2 - 13x + 3 = 0$

Form a quadratic equation where roots are 3/4 and 1/3, giving your answer in the form ax2 + bx + c = 0, whereas a, b, and c are integers.

Answer

Step-by-step solution

Identify the given roots.

Write the factors corresponding to these roots.

Form the quadratic equation by multiplying the factors.

Combine the $x$ terms and clear the denominators to get integer coefficients.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Form a quadratic equation where roots are 3/4 and 1/3, giving your answer in the form ax2 + bx + c = 0, whereas a, b, and c are integers.

Answer

Step-by-step solution

Identify the given roots.

Write the factors corresponding to these roots.

Form the quadratic equation by multiplying the factors.

Combine the xxx terms and clear the denominators to get integer coefficients.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Combine the $x$ terms and clear the denominators to get integer coefficients.