 Hong Kong
Stage 4 - Stage 5

# Solving 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.
• Factorise - fully factorising 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 factorise 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.