Solving quadratic equations

We always aim to have an equation with a quadratic expression on one side and zero on the other side.

Quadratics of the form \(ax^2+c\)

When \(b=0\), the easiest thing to do is to re-arrange the equation for \(x^2\) then take the square root. Do not forget the \(\pm\).

\begin{align} ax^2+b&=0 \\ x^2&=\frac{-b}{a} \\ x&=\pm\sqrt{\frac{-b}{a}} \end{align}

Notice that taking the square root of a negative number is not possible in real numbers. Therefore, equations of this form may not have solutions, specifically if \(\frac{-b}{a}\) is a negative number.

Quadratics of the form \(ax^2+bx\)

When \(c=0\), the easiest thing to do is to factor out \(x\) and use the zero-product rule: when the product of two things is zero, one of them must be zero.

\begin{align} ax^2+bx&=0 \\ x(ax+b)&=0 \end{align} \[\begin{array}{rlcrl} x&=0 &\quad\text{or}\quad & ax+b&=0 \\ x&=0 &\quad\text{or}\quad & x&=\frac{-b}{a} \end{array}\]

Equations of this form always have two solutions.

Quadratics of the form \(ax^2+bx+c\)

If the quadratic expression is factorable, we factor it then use the zero-product rule. For example,

\begin{align} 2x^2-5x-7&=0 \\ 2x^2-7x+2x-7&=0 \\ x(2x-7)+(2x-7)&=0 \\ (2x-7)(x+1)&=0 \end{align} \[\begin{array}{rlcrl} 2x-7&=0&\quad\text{or}\quad&x+1&=0 \\ x&=\frac{7}{2}&\quad\text{or}\quad&x&=-1 \end{array}\]

Do not forget the special pattern of perfect square.

If the quadratic is not factorable, then check the discrimnant \(\Delta\).

If \(\Delta\gt 0\) then there are two distinct (i.e. different) solutions.
If \(\Delta=0\) then there is a unique solution.
If \(\Delta\lt0\) then there is no real solution.

If there is a solution, use the quadratic formula to calculate them.