Ontario 10 Academic (MPM2D)
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Solve quadratics using various methods
Lesson

There are a number of ways to solve quadratics. Remember that when we say solve we are actually finding the $x$x-intercepts or roots of the equation.

We have seen:

  • Algebraically Solve - for simple binomial quadratics like $x^2=49$x2=49.
  • Factor - fully factoring a quadratic means we can then use the null factor law: if $a\times b=0$a×b=0 then either $a=0$a=0 or $b=0$b=0.
  • Completing the square - this method gets us to a point where we can then solve algebraically. It also tells us the vertex of the quadratic.
  • Quadratic Formula - this method will solve any quadratic function of the form $ax^2+bx+c=0$ax2+bx+c=0, but it is not always the easiest to deal with algebraically, sometimes the other methods are a better choice.

 

When looking to solve a quadratic, check for easy options:

  • Can we remove a common factor immediately? 
  • Can we solve it straight away algebraically?
  • Can we factor it easily?

If these first two options haven't worked then we can either complete the square or use the quadratic formula.

 

Let's have a look at these questions.

Question 1

Solve for $x$x:

$x^2=17x+60$x2=17x+60

  1. Write all solutions on the same line, separated by commas.

Question 2

Solve for $x$x, expressing your answer in exact form.

$\left(x-5\right)^2-4=8$(x5)24=8

  1. Write all solutions on the same line, separated by commas.

Question 3

Solve for the unknown:

$-8x+x^2=-6-x-x^2$8x+x2=6xx2

  1. Write all solutions on the same line, separated by commas.

Question 4

Solve the following equation:

$x-\frac{45}{x}=4$x45x=4

  1. Write all solutions on the same line, separated by commas.

Outcomes

10D.QR3.08

Solve quadratic equations that have real roots, using a variety of methods

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