Solving quadratics

Here's something that trips people up: a quadratic isn't just "an equation with x² in it" , it's specifically one that can be arranged into the form below, where the x² term is non-negotiable.

$a{x}^2+bx+c=0(a\ne 0)$

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant. The condition a ≠ 0 matters , if a were 0, the x² term would vanish and you'd just have a linear equation.

Quadratic Equation: Any equation that can be rearranged into $a{x}^2+bx+c=0$ where $a\ne 0$. The values that satisfy the equation are called the roots or zeros of the quadratic.

Think of solving a quadratic like cracking a combination lock that might have one, two, or even zero valid combinations , the discriminant (more on that later) tells you which situation you're in before you even start.

Every method for solving quadratics , factorisation, completing the square, the formula , assumes the equation is in standard form with zero on one side. Skip this step and you'll feed the wrong values of a, b, and c into your method, and your answer will be wrong even if your algebra is perfect.

Rearranging is just a case of moving every term to one side:

• $x^2=x+5\Rightarrow x^2-x-5=0$
• $2x-x^2=1\Rightarrow -x^2+2x-1=0$ (multiply through by −1 to get $x^2-2x+1=0$)
• $2m^2+23=14m\Rightarrow 2m^2-14m+23=0$