Quadratic Formula

linear-algebra

Definition

Quadratic Formula

The quadratic formula is a formula used to find the roots of a quadratic equation:

$$a{x}^2+bx+c=0$$

where a, b, c are coefficients and x is the variable.

The formula is given by:

$$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Derivation

The quadratic formula can easily be derived:

$$\begin{array}{rl}a{x}^2+bx+c& =0\\ x^2+\frac{b}{a}x+\frac{c}{a}& =0\\ x^2+\frac{b}{a}x& =-\frac{c}{a}\\ x^2+\frac{b}{a}x+{\left(\frac{b}{2a}\right)}^2-{\left(\frac{b}{2a}\right)}^2& =-\frac{c}{a}\\ x^2+\frac{b}{a}x+\frac{b^2}{4a^2}-\frac{b^2}{4a^2}& =-\frac{c}{a}\\ {\left(x+\frac{b}{2a}\right)}^2-\frac{b^2}{4a^2}& =-\frac{c}{a}\\ {\left(x+\frac{b}{2a}\right)}^2& =-\frac{c}{a}+\frac{b^2}{4a^2}\\ {\left(x+\frac{b}{2a}\right)}^2& =\frac{b^2-4ac}{4a^2}\\ x+\frac{b}{2a}& =\pm \sqrt{\frac{b^2-4ac}{4a^2}}\\ x+\frac{b}{2a}& =\pm \frac{\sqrt{b^2-4ac}}{\sqrt{4a^2}}\\ x+\frac{b}{2a}& =\pm \frac{\sqrt{b^2-4ac}}{2a}\\ x_{1,2}& =-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}\\ x_{1,2}& =\frac{-b\pm \sqrt{b^2-4ac}}{2a}\end{array}$$

Discriminant

The dicriminant of a quadratic equation defines the number of solutions of a quadratic function.

$$D=b^2-4ac$$

• $D\ge 0⟹$ has two distinct real roots
• $D=0⟹$ has one real root
• $D<0⟹$ has no real roots