## anonymous one year ago What is the sum of the roots of the equation listed below?

1. anonymous

$(x-\sqrt2)(x^2-\sqrt3x+\pi)=0$

2. anonymous

for a quadratic $$ax^2+bx+c$$ the sum of the roots is $$-b/a$$; observe: $$a(x-r_1)(x-r_2)=a(x^2-(r_1+r_2)x+r_1r_2)=ax^2\underbrace{-a(r_1+r_2)}_bx+\underbrace{ar_1r_2}_c\\\implies b =-a(r_1+r_2)\\\implies r_1+r_2=-\frac{b}a$$

3. anonymous

so the sum of the roots of $$x^2-\sqrt3x+\pi$$ is $$-(-\sqrt3)/1=\sqrt3$$, and then the root of $$x-\sqrt2$$ is clearly $$\sqrt2$$ so the total sum of our roots is $$\sqrt2+\sqrt3$$. this is because the roots of a product of two polynomials is simply the combined roots of both individual polynomials

4. anonymous

i see, thanks

5. Directrix

Sum of the Roots of a Polynomial Theorem Attached

6. anonymous

this chart is really helpful, thanks