2.2 Roots of Quadratic Equations

Roots of Quadratic Equations

Roots of a quadratic equation are the values of variables/unknowns that satisfy the equation.

Example:
Determine whether 1, 2, and 3 are the roots of the quadratic equation $x^2-<!--\; -\; -->5x+6=0$ .

Answer:
When x = 1,
$\begin{array}{l}{x}^2-<!--\; -\; -->5x+6=0\\ (1{)}^2-<!--\; -\; -->5(1)+6=0\\ 2=0\end{array}$
x = 1 does not satisfy the equation

When x = 2,
$\begin{array}{l}{x}^2-<!--\; -\; -->5x+6=0\\ (2{)}^2-<!--\; -\; -->5(2)+6=0\\ 0=0\end{array}$
x = 2 satisfies the equation.

When x = 3
$\begin{array}{l}{x}^2-<!--\; -\; -->5x+6=0\\ (3{)}^2-<!--\; -\; -->5(3)+6=0\\ 0=0\end{array}$
x = 3 satisfies the equation.

 Conclusion:

1. 2 and 3 satisfy the equation $x^2-<!--\; -\; -->5x+6=0$ , hence there are the roots of the equation.
2. 1 does not satisfy the equation $x^2-<!--\; -\; -->5x+6=0$ , hence it is NOT the root of the equation.

Remember : Roots of a quadratic equation are the values of variables/unknowns that satisfy the equation.

Example
(a) Given x = 3 is the root of the quadratic equation $x^2+2x+p=0$ , find the value of p.
(b) The roots of the quadratic equation $3x^2+hx+k=0$   are -2 and 4. Find the value of h and k.