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A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).

Important!

The equation \(ax^2 + bx + c = 0\) is called the standard form of a quadratic equation.

Roots of a quadratic equation

The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.

Consider the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\)

Divide the equation by \(a\).

${x}^{2}+\frac{b}{a}x+\frac{c}{a}=0$

Move the constant to the right side.

${x}^{2}+\frac{b}{a}x=-\frac{c}{a}$

Add the square of one-half of the coefficient of \(x\) on both sides.

${x}^{2}+\frac{b}{a}x+{\left(\frac{b}{2a}\right)}^{2}=-\frac{c}{a}+{\left(\frac{b}{2a}\right)}^{2}$

${\left(x+\frac{b}{2a}\right)}^{2}=-\frac{c}{a}+\frac{{b}^{2}}{4{a}^{2}}$

${\left(x+\frac{b}{2a}\right)}^{2}=\frac{{b}^{2}-4\mathit{ac}}{4{a}^{2}}$

Taking square root on both sides.

$x+\frac{b}{2a}=\pm \sqrt{\frac{{b}^{2}-4\mathit{ac}}{4{a}^{2}}}$

$x+\frac{b}{2a}=\pm \frac{\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

$x=-\frac{b}{2a}\pm \frac{\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

$x=\frac{-b\pm \sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

Therefore, the roots of \(ax^2 + bx + c = 0\) are $x=\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ and $x=\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$.