5 Shortcuts to Solve Quadratic Equations Faster

Here are 5 shortcuts to solve quadratic equations faster:

1. Square Root Method (When $b=0$)

If the equation is in the form $a{x}^2+c=0$, solve as:

$$x=\pm \sqrt{-\frac{c}{a}}$$

Example:
$2x^2-8=0$
$x^2=4$
$x=\pm 2$

2. Factor Out Common Terms

If all terms have a common factor, divide first to simplify.
Example:
$3x^2+6x+3=0$
Divide by 3:
$x^2+2x+1=0$
$(x+1{)}^2=0$
$x=-1$

3. Reverse FOIL (Factoring)

If the quadratic can be factored easily, use:
$x^2+(a+b)x+ab=(x+a)(x+b)$.
Example:
$x^2-5x+6=0$
Factors: $(x-2)(x-3)=0$
Solutions: $x=2,3$

4. Sum & Product Shortcut (For Monic Quadratics)

For $x^2+bx+c=0$, find two numbers that:

• Add to $b$

• Multiply to $c$
  Example:
  $x^2-4x-12=0$
  Numbers: $-6$ and $+2$ (since $-6+2=-4$ and $-6\times 2=-12$)
  Roots: $x=6,-2$

5. Quick Quadratic Formula (When $a=1$)

For $x^2+bx+c=0$, use:

$$x=\frac{-b\pm \sqrt{b^2-4c}}{2}$$

(Simplified version of the standard formula.)
Example:
$x^2-6x+8=0$
$x=\frac{6\pm \sqrt{36-32}}{2}=\frac{6\pm 2}{2}$
Solutions: $x=4,2$

Bonus: Vertex Shortcut (For Graphing)

The vertex of $y=a{x}^2+bx+c$ is at $x=-\frac{b}{2a}$.
Useful for finding maxima/minima quickly.

These shortcuts save time when applicable! Always check if the quadratic fits these special cases before using the full quadratic formula.