The quadratic formula

Another way of solving a quadratic equation on the form of

$$ax^{2}+bx+c=0$$

Is to used the quadratic formula. It tells us that the solutions of the quadratic equation are

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

$$\: \: where\: \: a\neq 0\: \: and\: \: b^{2}-4ac\geq 0$$

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Example

Solve the equation

$$x^{2}{\color{green} {\, -\, 3}}x{\color{blue} {\, -\, 10}}=0$$

$$x=\frac{-\left ( {\color{green}{ -\, 3}} \right )\pm \sqrt{\left ( {\color{green} {-\, 3}} \right )^{2}-4\cdot \left ( 1\cdot {\color{blue} {-\, 10}} \right )}}{1\cdot 2}$$

$$x=\frac{3\pm \sqrt{9+40}}{2}$$

$${\color{red}{ x_{1}}}=\frac{3{\color{red}{ \, +\, }\sqrt{49}}}{2}=  \frac{3+7}{2}= \frac{10}{2}= {\color{red} {5}}$$

$${\color{red} {x_{2}}}=  \frac{3{\color{red}{ \, -\, }\sqrt{49}}}{2}= \frac{3-7}{2}= \frac{-4}{2}= {\color{red} {-2}}$$

The expression

$$b^{2}-4ac$$

Within the quadratic formula is called the discriminant. The discriminant can be used to determine how many solutions the quadratic equation has.

$$\begin{matrix} if\: \: b^{2}-4ac>0 & &\: \: \: \: \: \: \: \: \: \: 2\: \: solutions \\ if\: \: b^{2}-4ac=0 & & \: \: \: \: \: \: \: \: \: \: \: \: 1\: \: solution\\ if\: \: b^{2}-4ac<0 & & no\: \: real\: \: solution \end{matrix}$$

Here you can check that you've got the right solution

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Video lesson

Solve the equation using the quadratic formula

$$x^{2} - 3x-10$$