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The quadratic formula

The Quadratic Formula

The quadratic formula allows the solutions of an equation of the form $ax^2 + bx + c =0$ to be determined exactly. This formula works for $a,b,c,x\in \mathbb{C}$, that is, the formula can work for both real and complex numbers. The formula is $$ x = \frac{-b\pm\sqrt{b^2- 4ac}}{2a} $$ The expression $b^2 - 4ac$ is referred to as the discriminant and has the following properties: Given $a,b,c\in\mathbb{R}$, then
  • $b^2-4ac \gt 0$ then 2 real solutions
  • $b^2-4ac = 0$ then 1 real solution
  • $b^2-4ac \lt 0$ then 0 real solutions

Quadratic formula 1

Solve the equation $2x^2 + 4x - 3 = 0$
solution - press button to display
The coefficients $a,b,c$ are $a=2$, $b=4$, $c=-3$ The quadratic formula is $$ x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $$ Substitution of the coefficients gives $$ x = \frac{-4\pm\sqrt{4^2 - 4(2)(-3)}}{2\times 2} $$ Simplification gives $$ x = \frac{-4\pm \sqrt{16 + 24}}{4} = \frac{-2\pm \sqrt{10}}{2} $$