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Stage 5.1-3

4.07 Further quadratic equations

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Further quadratic equations

We have looked at a number of ways to solve  quadratic equations  . These include:

  •  Factorising  the quadratic expression and then using the null factor law. In particular, when the quadratic expression is non-monic, we can use techniques for non-monic quadratic trinomials.

  • Completing the square and then taking the square root of both sides of the equation.

  • Using the  quadratic formula  .

Examples

Example 1

Solve 5x^{2}+22x+8=0 for x by factorising or otherwise.

Worked Solution
Create a strategy

Factorise the equation and then solve for x.

Apply the idea
\displaystyle 5x^{2}+22x+8\displaystyle =\displaystyle 0Write the equation
\displaystyle 5x^{2}+2x+20x+8\displaystyle =\displaystyle 0Split 22x
\displaystyle x(5x+2)+4(5x+2)\displaystyle =\displaystyle 0Factor each pair of terms
\displaystyle (5x+2)(x+4)\displaystyle =\displaystyle 0Factor out 5x+2
\displaystyle 5x+2\displaystyle =\displaystyle 0Equate the first factor to 0
\displaystyle 5x\displaystyle =\displaystyle -2Subtract 2 from both sides
\displaystyle x\displaystyle =\displaystyle -\dfrac{2}{5}Divide both sides by 5
\displaystyle x+4\displaystyle =\displaystyle 0Equate the second factor to 0
\displaystyle x\displaystyle =\displaystyle -4Subtract 4 from both sides

So the solutions are x=-\dfrac{2}{5},\,x=-4.

Example 2

Solve 4x^{2}+11x+7=0 for x by completing the square or otherwise.

Worked Solution
Create a strategy

Use completing the square.

Apply the idea
\displaystyle 4x^{2}+11x+7\displaystyle =\displaystyle 0Write the equation
\displaystyle x^{2}+\dfrac{11}{4}x + \dfrac{7}{4}\displaystyle =\displaystyle 0Divide both sides by 4
\displaystyle x^{2}+\dfrac{11}{4}x\displaystyle =\displaystyle -\dfrac{7}{4}Subtract \dfrac{7}{4} from both sides
\displaystyle x^{2}+\dfrac{11}{4}x+\left(\dfrac{11}{8}\right)^{2}\displaystyle =\displaystyle \left(\dfrac{11}{8}\right)^{2}-\dfrac{7}{4}Add half of \dfrac{11}{4} squared to both sides
\displaystyle \left(x+\dfrac{11}{8}\right)^{2}\displaystyle =\displaystyle \left(\dfrac{11}{8}\right)^{2}-\dfrac{7}{4}Write the left side as a perfect square
\displaystyle \left(x+\dfrac{11}{8}\right)^{2}\displaystyle =\displaystyle \dfrac{9}{64}Evaluate the right side
\displaystyle x+\dfrac{11}{8}\displaystyle =\displaystyle \pm\sqrt{\dfrac{9}{64}}Take the square root of both sides
\displaystyle x+\dfrac{11}{8}\displaystyle =\displaystyle \pm\dfrac{3}{8}Evaluate the square root
\displaystyle x\displaystyle =\displaystyle -\dfrac{11}{8} \pm \dfrac{3}{8}Subtract \dfrac{11}{8} from both sides
\displaystyle =\displaystyle -1,\,-\dfrac{7}{4}Evaluate for each sign and simplify

So the solutions are x=-1,\,x=-\dfrac{7}{4}.

Example 3

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

Worked Solution
Create a strategy

We can use the quadratic formula: x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}.

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{3}{4}Evaluate
\displaystyle x\displaystyle =\displaystyle \dfrac{-7-1}{8} Take the negative square root
\displaystyle =\displaystyle -1Evaluate and simplify

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

Idea summary

Ways to solve quadratic equations:

  • Factorising the quadratic expression and then using the null factor law.

  • Completing the square.

  • Using the quadratic formula.

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

MA5.3-9NA

sketches and interprets a variety of non-linear relationships

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