The Quadratic Formula

The Quadratic Formula

The quadratic formula solves any quadratic written as $ax^2+bx+c=0$: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. It is especially useful when factorisation is difficult. Substitute values with brackets so negative coefficients are protected, calculate the discriminant carefully, and write both roots or explain why there are no real roots.

Worked reasoning

For $2x^2-3x-2=0$, $a=2$, $b=-3$, $c=-2$. Then $x=\frac{3\pm\sqrt{9+16}}4=\frac{3\pm5}4$, giving $2$ and $-1/2$.

Exam method

1. Rearrange to zero. 2. Identify $a$, $b$, $c$ with signs. 3. Substitute into the formula, simplify, and check one root if time allows.

Formula-substitution routine

Use a three-line coefficient table before pressing a calculator key: $a=$, $b=$, $c=$. This prevents the most expensive error in Grade 11 quadratics: losing a negative sign during substitution. Use brackets around every coefficient in the formula, especially $b$ and $c$. If a final root looks implausible, substitute it into the original quadratic. Even a quick check of one root can reveal whether a copied sign or denominator is wrong.

One-minute retrieval: The Quadratic Formula

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.