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
Question 2
Solve for $p$p:
$5\left(p^2-3\right)=705$5(p2−3)=705
Question 3
Solve for $x$x:
$\left(4x+3\right)^2=64$(4x+3)2=64
Write all solutions on the same line, separated by commas.
Factorisation and null factor law
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 4
Solve $x^2+6x-55=0$x2+6x−55=0 for $x$x.
Write all solutions on the same line, separated by commas.
Question 5
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 6
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.
Solving quadratic equations using CAS calculator
Your CAS calculator has a number of ways to solve equations. One method is to graph both sides of the equation and look for the point of intersection. Most graphic calculators also have a solve function. For example: selecting Action ► Equation/Inequality ► solve to create the instruction solve(3x2 - 4 = 12) in the main window of the Casio classpad will generate the two solutions to the equation $3x^2-4=12$3x2−4=12. Note that it's often easiest to use the variable $x$x in the equation, even if the given equation isn't written with $x$x as the variable.
Experiment with your calculator and ensure you are able to solve the equation in the practice question below using your calculator.
Practice question
Question 7
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.