Question

Write a quadratic equation with the given roots. Write the equation in the form ax^2+bx+c=0 where a, b, and c are integers.

Answer 1

We have the roots 1/2 and 4/3 and we need to write a quadratic equation of the form $a x^{2} +bx+c$ using those roots. Since we know that 1/2 and 4/3 are roots of our quadratic, we can equate each root to $x$ and factor backwards: 
$x= \frac{1}{2}$ and $x= \frac{4}{3}$
$2x=1$ and $3x=4$
$2x-1=0$ and $3x-4=0$

Now we can multiply the left sides of our equations to get our quadratic:
$(2x-1)(3x-4)=6 x^{2} -8x-3x+4=6 x^{2} -11x+4$

W can conclude that the quadratic equation with roots 1/2 and 4/3 is $6x^2-11x+4$.