We have the roots 1/2 and 4/3 and we need to write a quadratic equation of the form [tex]a x^{2} +bx+c[/tex] using those roots. Since we know that 1/2 and 4/3 are roots of our quadratic, we can equate each root to [tex]x[/tex] and factor backwards: [tex]x= \frac{1}{2} [/tex] and [tex]x= \frac{4}{3} [/tex] [tex]2x=1[/tex] and [tex]3x=4[/tex] [tex]2x-1=0[/tex] and [tex]3x-4=0[/tex]
Now we can multiply the left sides of our equations to get our quadratic: [tex](2x-1)(3x-4)=6 x^{2} -8x-3x+4=6 x^{2} -11x+4[/tex]
W can conclude that the quadratic equation with roots 1/2 and 4/3 is [tex]6x^2-11x+4[/tex].
