There are many methods to solve quadratic equations including:
- Algebraic manipulation
- Factorisation followed by the null factor law
- Technology
- Completing the square
- The quadratic formula
Let's look at the first three methods in more detail. The last two methods will be covered in the lessons that follow.
Algebraic manipulation
Solving some quadratic equations can be achieved by isolating a variable using inverse operations.
Practice questions
Question 1
Question 2
Solve the equation $2\left(x-3\right)^2=8$2(x−3)2=8.
Write all solutions on the same line, separated by commas.
Factorisation
There are many techniques for factorising quadratics, and these are covered in the preceding exercise.
There is a great benefit to factorising quadratics in order to solve them. Once a quadratic is factorised, the null factor law can be used to find the values of $x$x that solve the equation.
The null factor law states that if $a\times b=0$a×b=0 then $a=0$a=0 or $b=0$b=0. This law is very useful when solving factorised polynomials because, by writing a quadratic in the factored form $a(x)\times b(x)=0$a(x)×b(x)=0 where $a(x)$a(x) and $b(x)$b(x) are linear factors, setting each factor equal to zero and rearranging to solve for the variable in each will give a solution.
If the product of any two terms is equal to zero, then one or both of these terms equals zero.
That is, if $a\times b=0$a×b=0 then $a=0$a=0 or $b=0$b=0 . This also includes the possibility that both terms are equal to zero.
The following examples use different factorising techniques. Note that the method of solving, once factorised, is always the same.
Worked example
example 1
Solve $2x^2+12x=0$2x2+12x=0.
Think: Notice that both of these terms have a common factor of $2x$2x, so we can factorise this equation using common factors.
Do :
| $2x^2+12x$2x2+12x | $=$= | $0$0 |
| $2x\left(x+6\right)$2x(x+6) | $=$= | $0$0 |
So either
| $2x$2x | $=$= | $0$0 | or | $\left(x+6\right)$(x+6) | $=$= | $0$0 |
| $x$x | $=$= | $0$0 | $x$x | $=$= | $-6$−6 |
Practice questions
Question 3
Solve $x^2+6x-55=0$x2+6x−55=0 for $x$x.
Write all solutions on the same line, separated by commas.
Question 4
Solve the following equation by first factorising the left hand side of the equation.
$5x^2+22x+8=0$5x2+22x+8=0
Write all solutions on the same line, separated by commas.
Question 5
Solve the following equation for $b$b using the PSF method of factorisation: $15-11b-12b^2=0$15−11b−12b2=0
Write all solutions in fraction form, on the same line separated by commas.
Using technology
Up to this point, different algebraic techniques have been used as ways to solve quadratic equations. A range of these methods are algebraic, meaning that they focus on manipulation of the algebraic equation to find the solutions.
For some quadratic equations, the solution may be found by graphing the function of the quadratic. However, with many quadratics, it is not possible to be consistent and neat enough when graphing by hand to read off the vertex and intercepts of a parabola accurately.
Luckily, there are many forms of technology available today that can help solve equations both algebraically and graphically. The great thing about using computers when exploring mathematics is that, once the techniques & concepts are understood, they can then be used to help complete long or repetitive calculations effectively.
Practice question
Question 6
Using the solve command on your calculator, or otherwise, find the roots of $4.6x^2+7.3x-3.7=0$4.6x2+7.3x−3.7=0.
Give your answers as decimal approximations to the nearest tenth. Write the decimal approximations for both roots on the same line, separated by a comma.
Modelling situations using quadratics
The methods explored so far can also allow solutions to real-life situations to be found, providing they are able to be modelled by a quadratic equation. Remember to consider the context of the situation when providing a final answer, as some solutions might not apply. For example, if the variable in a quadratic equation represents time, any negative solutions would not make sense in context.
Practice question
Question 7
Homer needs a sheet of paper $x$x cm by $32$32 cm for an origami koala. The local origami supply store only sells square sheets of paper.
The lower portion of the image below shows the excess area $A$A of paper that will be left after Homer cuts out the $x$x cm by $32$32 cm piece. The excess area, in cm2, is given by the equation $A=x\left(x-32\right)$A=x(x−32).
At what lengths $x$x will the excess area be zero?
Write all solutions on the same line, separated by a comma.
For what value of $x$x will Homer be able to make an origami koala with the least amount of excess paper?