Sum and product of the roots

Statement

An equation $x^2+bx+c=0$ has roots whose sum is $7$ and whose product is $10$.

• (a) Find $b$ and $c$ and the roots.
• (b) Write the equation whose roots are the original ones increased by $2$.

Solution

(a) By the relations between roots and coefficients: $x_1+x_2=-b=7\Rightarrow b=-7$; $x_1 x_2=c=10$. The equation is $x^2-7x+10=0=(x-2)(x-5)$, with roots $x_1=2$, $x_2=5$.

(b) The new roots are $4$ and $7$: sum $11$, product $28$, so $x^2-11x+28=0$.

$\boxed{b=-7,\ c=10,\ \text{roots } 2\ \text{and}\ 5;\quad \text{new eq. } x^2-11x+28=0}$