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3 Quadratic equations
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Stage 5.1-2

3.02 Solving quadratic equations by factorisation

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Lesson

Quadratic equation by factorisation

In the last chapter we looked at how to solve quadratic equations using  the null factor law  . This gives us a method for solving most quadratic equations:

  1. Gather together all the terms on one side of the equation so that the other side is 0.

  2.  Factorise  these terms using any appropriate techniques.

  3. Use the null factor law to split the quadratic equation into two linear equations.

  4. Solve the linear equations to get the solutions to the original quadratic equation.

Examples

Example 1

The equation 3y-15y^{2}=0 has two solutions. Solve for both values of y.

Worked Solution
Create a strategy

Factorise the expression on the left and use the null factor law to solve.

Apply the idea
\displaystyle 3y-15y^{2}\displaystyle =\displaystyle 0Write the equation
\displaystyle 3y(1-5y)\displaystyle =\displaystyle 0Factorise the quadratic expression
\displaystyle 3y\displaystyle =\displaystyle 0Equate the first factor to 0
\displaystyle y\displaystyle =\displaystyle 0Divide both sides by 3
\displaystyle 1-5y\displaystyle =\displaystyle 0Equate the second factor to 0
\displaystyle 5y\displaystyle =\displaystyle 1Add 1 to both sides
\displaystyle y\displaystyle =\displaystyle \dfrac{1}{5}Divide both sides by 5

So the solutions are y=0 and y=\dfrac{1}{5}.

Example 2

Solve x^{2}=17x+60 for x.

Worked Solution
Create a strategy

Move all terms to the left then factorise and use the null factor law to solve.

Apply the idea
\displaystyle x^{2}\displaystyle =\displaystyle 17x+60Write the equation
\displaystyle x^{2}-17x-60\displaystyle =\displaystyle 0Move all the terms to the left hand side
\displaystyle (x-20)(x+3)\displaystyle =\displaystyle 0Factorise the quadratic expression
\displaystyle x-20\displaystyle =\displaystyle 0Equate the first factor to 0
\displaystyle x\displaystyle =\displaystyle 20Add 20 to both sides
\displaystyle x+3\displaystyle =\displaystyle 0Equate the second factor to 0
\displaystyle x\displaystyle =\displaystyle -3Subtract 3 from both sides

So the solutions are x=20 and x=-3.

Idea summary

We can solve most quadratic equations by using this method:

  1. Gather together all the terms on one side of the equation so that the other side is 0.

  2. Factorise these terms using any appropriate techniques.

  3. Use the null factor law to split the quadratic equation into two linear equations.

  4. Solve the linear equations to get the solutions to the original quadratic equation.

Outcomes

MA5.2-8NA

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

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