A quadratic equation in one variable has the form $ax^2+bx+c=0$ where $a\neq0$. A root is a value of $x$ that makes the equation true.
Solving by factorisation
Write the equation as a product of two factors equal to zero, then use the fact that if $PQ=0$ then $P=0$ or $Q=0$. For example $x^2-5x+6=0$ becomes $(x-2)(x-3)=0$, giving $x=2$ or $x=3$.
Key formula
Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Sum of roots $=-\frac{b}{a}$, product of roots $=\frac{c}{a}$.
The discriminant
The value $b^2-4ac$ is the discriminant. It tells us the number of real roots without solving: two distinct real roots when $b^2-4ac>0$, one repeated root when $b^2-4ac=0$, and no real roots when $b^2-4ac<0$.
Worked example
Solve $2x^2-5x-3=0$. Here $a=2,b=-5,c=-3$, so $b^2-4ac=25+24=49>0$ (two roots). Then $x=\frac{5\pm\sqrt{49}}{4}=\frac{5\pm7}{4}$, giving $x=3$ or $x=-\frac{1}{2}$.
Remember
- Always rearrange to $ax^2+bx+c=0$ first.
- Check the signs of $b$ and $c$ when reading off $a,b,c$.
- Sum $=-b/a$ and product $=c/a$ help you verify roots.