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CanadaON
Grade 9

4.05 Simple quadratic equations

Lesson

We call equations like $$x7=2 linear equations. These are equations where all variables have a power of $$1. Linear equations have only one solution. That is, there is only one value of the variable which will make the equation true. In this case, the only solution is $$x=9

If instead, the variables in the equation have a power of $$2, we call them quadratic equations. Quadratic equations can potentially have two solutions. For example, in the equation $$x27=2, there are two solutions, $$x=3 and $$x=3.

Exploration

Solve $$x2+4=20 for $$x.

Following our rules for solving linear equations, we want to isolate $$x and whatever we do to one side of the equation we do to the other. So our first step is to subtract $$4 from both sides of the equation, giving us $$x2=16.

Next we want to undo raising $$x to the power of $$2. However, we need to be careful here. There are two operations which could be the reverse of squaring a number. We have to take both the positive and negative square root. This will give us two solutions.

$$x2 $$= $$16  
$$x $$= $$±16

Taking the positive and negative square root of both sides

  $$= $$±4

Evaluating the positive and negative square roots

The symbol $$± means "plus or minus". We can use this as a shorthand for both the positive and negative of a number. In this case, it means that our solutions are $$x=4 and $$x=4.

We can check these solutions by substituting them in to the original equation and seeing if it holds true.

 

 

Did you know?

We call raising a number to the power of $$2 "squaring" the number. "Quadratic" comes from the ancient Latin for "square". So quadratic equations can be thought of as square equations.

Summary

Equations where all of the variables have a power of one are linear equations. These have at most one solution.

Equations where some of the variables have a power of two are quadratic equations. These have at most two solutions.

The symbol $$± means "plus or minus".

If we can rearrange a quadratic equation into the form $$x2=k, then we can solve the equation by taking the positive and negative square roots. That is, $$x=±k.

Practice questions

Question 1

Solve $$x2=2 for $$x.

Enter each solution as a radical on the same line, separated by a comma.

Question 2

Solve $$x2162=2 for $$x.

Enter each solution on the same line, separated by a comma.

 
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