Respuesta :
You haven't provided the required roots, but I can tell you how to do this kind of exercises in general.
If the [tex] x^2 [/tex] coefficient is 1, i.e. the equation is written like [tex] x^2+bx+c=0 [/tex], then you can say the following about the coefficients b and c:
- [tex] b [/tex] is the opposite of the sum of the roots
- [tex] c [/tex] is the multiplication of the roots.
So, for example, if we want an equation whose roots are 4 and -2, we have:
- [tex] 4+(-2) = 4-2 = 2 \implies b = -2 [/tex]
- [tex] 4 \cdot (-2) = -8 \implies c = -8 [/tex]
So, the equation is [tex] x^2-2x-8=0 [/tex]
If your roots are rational, you can work like this: suppose you want an equation with roots 3/4 and 1/2. You have:
- [tex] \dfrac{3}{4}+\dfrac{1}{2} = \dfrac{3}{4}+\dfrac{2}{4} = \dfrac{5}{4} \implies b = -\dfrac{5}{4} [/tex]
- [tex] \dfrac{3}{4} \cdot \dfrac{1}{2} = \dfrac{3}{8} \implies c = \dfrac{3}{8} [/tex]
And so the equation is
[tex] x^2 - \dfrac{5}{4} + \dfrac{3}{8} = 0 [/tex]
In order to have integer coefficients, you can multiply both sides of the equation by 8:
[tex] 8x^2 - 10 + 3 = 0 [/tex]
