Q.Verify whether $x = -1$ is a root of the equation $2x^2 + 5x + 3 = 0$. If it is, find the other root using the sum-of-roots relation.

First substitute $x=-1$ to check it is a root, then use the sum-of-roots relation $-b/a$ to find the other root without re-solving the whole equation.

Let $p(x)=2x^2+5x+3$. Substitute $x=-1$: $p(-1)=2(-1)^2+5(-1)+3=2(1)-5+3=2-5+3=0$.

Since $p(-1)=0$, $x=-1$ is a root of the equation.

To find the other root without redoing the full solve, use the sum-of-roots relation for $ax^2+bx+c=0$: $x_1+x_2=-\dfrac{b}{a}$. Here $a=2, b=5$, so $x_1+x_2=-\dfrac{5}{2}$.

With $x_1=-1$ known: $x_2 = -\dfrac52 - (-1) = -\dfrac52+1 = -\dfrac32$. …

Alternative — factorization directly: $ac=2\times3=6$; need $p,q$ with product $6$, sum $5$: $p=2,q=3$. $2x^2+2x+3x+3=0\Rightarrow 2x(x+1)+3(x+1)=0\Rightarr …

A common error is using the product-of-roots relation ($c/a$) instead of the sum-of-roots relation, or mixing up their formulas — remember $x_1+x_2=-b/a$ (note the minus sign) while $x_1 x_2=c/a$ (no minus sign). Confusing w …