|
The roots of the quadratic equation
are given by the formula $$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. $$
The number $\Delta=b^2-4ac$ is called the discriminant of the equation. If $\Delta>0$ , there are two different real roots, if $\Delta=0$ there is a single real root, and if $\Delta<0$ there are no real roots (but two different complex roots).
Let's work a few examples.
First, consider $2x^2-14x+24=0$ . Here $a=2$ , $b=-14$ , and $c=24$ . Substituting in the formula gives us $$ x=\frac{14\pm \sqrt{(-14)^2-4\cdot2\cdot24}}{2\cdot 2} =\frac{14\pm\sqrt{4}}{4} =\frac{14\pm2}{4} =\frac{7\pm1}{2}. $$ So we have two solutions (depending on whether we take the sign $+$ or $-$ ): $x=\frac{8}{2}=4$ and $x=\frac{6}{2}=3$ .
Now we will solve $x^2-x-1=0$ . Here $a=1$ , $b=-1$ , and $c=-1$ , so $$ x=\frac{1\pm\sqrt{(-1)^2-4(1)(-1)}}{2} =\frac{1\pm{\sqrt{5}}}{2}, $$ and the solutions are $x=\frac{1+\sqrt{5}}{2}$ and $x=\frac{1-\sqrt{5}}{2}$ .
| 
