Quadratic Equations

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 $x^2=49$`x`2=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 $ax^2+bx+c=0$`a``x`2+`b``x`+`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.

Solve for $x$`x`:

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

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

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.

Solve for the unknown:

$-8x+x^2=-6-x-x^2$−8`x`+`x`2=−6−`x`−`x`2

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

Solve the following equation:

$x-\frac{45}{x}=4$`x`−45`x`=4

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

Manipulate rational, exponential, and logarithmic algebraic expressions

Form and use linear, quadratic, and simple trigonometric equations

Apply algebraic methods in solving problems