Solve quadratics using various methods
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
Write all solutions on the same line, separated by commas.
Question 2
Solve for $x$x, expressing your answer in exact form.
$\left(x-5\right)^2-4=8$(x−5)2−4=8
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=−6−x−x2
Write all solutions on the same line, separated by commas.
Question 4
Solve the following equation:
$x-\frac{45}{x}=4$x−45x=4
Write all solutions on the same line, separated by commas.