The quadratic formula gives the solutions to any quadratic equation in one variable. It can be used to solve quadratic equations which cannot be solved by factorisation.

Before we can use the quadratic formula, we have to rearrange the quadratic equation into the form ax^{2}+bx+c=0, where a,\,b,\, and c are any number and a \neq 0. Once the equation is in this form, the solutions are given by the quadratic formula: x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}

Exploration

The following applet shows the graphical representation of solutions to a quadratic equation. Try moving the sliders for A,\,B and C and see what happens. What do you notice?

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When a is positive the parabola opens upwards, when a is negative the parabola opens downwards. The values of b and c also affect the shape of the parabola.

Examples

Example 1

Solve 4x^{2}+7x+3 =0 for x by using the quadratic formula or otherwise.

Worked Solution

Create a strategy

Use the quadratic formula.

Apply the idea

\displaystyle x	\displaystyle =	\displaystyle \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}	Write the quadratic formula
\displaystyle x	\displaystyle =	\displaystyle \dfrac{-7\pm\sqrt{7^{2}-4\times 4 \times 3}}{2\times 4}	Substitute a=4,\, b=7, \,c=3
	\displaystyle =	\displaystyle \dfrac{-7\pm\sqrt{1}}{8}	Evaluate the square root
\displaystyle x	\displaystyle =	\displaystyle \dfrac{-7+1}{8}	Take the positive square root
	\displaystyle =	\displaystyle \dfrac{-6}{8}	Evaluate the numerator
	\displaystyle =	\displaystyle -\dfrac{3}{4}	Simplify
\displaystyle x	\displaystyle =	\displaystyle \dfrac{-7-1}{8}	Take the negative square root
	\displaystyle =	\displaystyle \dfrac{-8}{8}	Evaluate the numerator
	\displaystyle =	\displaystyle -1	Simplify

So the solutions are x = -\dfrac{3}{4} and x=-1.

Example 2

Solve 10-6m+2m^{2}=m^{2}+8m+9 for m by using the quadratic formula or otherwise.

Worked Solution

Create a strategy

Rearrange the equation into the form ax^{2}+bx+c=0, and use the quadratic formula.

Apply the idea

\displaystyle 10-6m+2m^{2}	\displaystyle =	\displaystyle m^{2}+8m+9	Write the equation
\displaystyle m^{2}-14m+1	\displaystyle =	\displaystyle 0	Subtract m^2, \, 8m, \,9 from both sides
\displaystyle m	\displaystyle =	\displaystyle \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}	Write the quadratic formula
\displaystyle m	\displaystyle =	\displaystyle \dfrac{-(-14)\pm\sqrt{(-14)^{2}-4\times 1 \times 1}}{2\times 1}	Substitute a=1,\, b=-14, \,c=1
	\displaystyle =	\displaystyle \dfrac{14\pm\sqrt{192}}{2}	Evaluate the square root
	\displaystyle =	\displaystyle \dfrac{14\pm 8\sqrt{3}}{2}	Simplify the surd
	\displaystyle =	\displaystyle 7\pm 4\sqrt{3}	Simplify the fraction

So the solutions are m=7+4\sqrt{3} and m=7-4\sqrt{3}.

Idea summary

For a quadratic equation of the form ax^{2}+bx+c=0, the solutions are given by the quadratic formula: x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}

Outcomes

MA5.2-8NA

solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

MA5.3-7NA

solves complex linear, quadratic, simple cubic and simultaneous equations, and rearranges literal equations

4.03 The quadratic formula

Ideas

The quadratic formula