What is quadratic equation?

A quadratic equation is an expression like the following

\[ ax^{2}+bx+c=0\,\left(a\neq0\right)\notag \]

The reason for \(a\neq 0\) is that if \(a=0\), the \(x^2\) term disappears and the order of the equation becomes less than first order.

What is solution formula?

Quadratic equations have solution formula.

The solution formula can be divided into the following cases by the discriminant (\( D=b^2-4ac \))

\[ x= \begin{cases} \displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}&\left( D\gt0 \right)\\[+10pt] \displaystyle -\frac{b}{2a}&\left( D=0 \right)\\[+10pt] \displaystyle \frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}&\left( D\lt0 \right) \end{cases} \]

Example

Example 1

Find the solution of the following quadratic equation.

\[ x^{2}-4x+4=0 \]

Solution

The value of discriminant \(D\) can be calculated as follows

\[ D=\left(-4\right)^{2}-4\cdot1\cdot4=0 \]

Therefore, from the solution formula for \(D=0\), the solution can be calculated as follows.

\[ x=-\frac{-4}{2\cdot1}=2 \]

Example 2

Find the solution of the following quadratic equation.

\[ x^{2}+x+1=0 \]

Solution

The value of discriminant \(D\) can be calculated as follows

\[ D=1^{2}-4\cdot 1\cdot 1=-3 \]

Therefore, from the solution formula for \(D\lt0\), the solution can be calculated as follows.

\begin{align} x&=\frac{-1\pm\sqrt{4\cdot 1\cdot1-1^{2}}}{2\cdot 1}\notag\\[+5pt] &=\frac{-1\pm\sqrt{3}}{2}\notag \end{align}