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 quadratics can be achieved by using inverse operations. Let's look at a few examples of this process.
Practice questions
Question 1
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, these are covered in the preceding exercise.
There is a great benefit to factorising quadratics in order to solve them. Once we factorise the quadratic, we can make use of the null factor law to find the values of $x$x that solve the equation.
Remember: If $a\times b=0$a×b=0 then $a=0$a=0 or $b=0$b=0. This is known as the null factor law.
Let's have a look at some examples using 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 these terms have a common factor of $2x$2x, so we can factorise this one 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 2
Solve $x^2+6x-55=0$x2+6x−55=0 for $x$x.
Write all solutions on the same line, separated by commas.
Question 3
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 4
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, we have looked at many different ways to solve quadratic equations. A range of these methods are algebraic, meaning we focus on manipulation of the algebraic equation to find the solutions.
For some quadratic equations, we may be able to find the solution 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 us to solve equations both algebraically and graphically. The great thing about using computers when exploring mathematics is that, once we understand and are confident with the concepts, we can use them to help us complete long or repetitive calculations effectively.
The question below uses a CAS calculator to solve a quadratic. However, you could also use websites like Desmos or WolframAlpha's quadratic formula calculator to solve quadratic equations using technology.
Practice question
Question 5
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.