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Quadratic equations Quadratic formula Factorising quadratics Simplifying expressions Completing the square Expanding bracketsThis topic is relevant for:
Here we will learn about solving quadratic equations by factorising including how to solve quadratic equations by factorising when
There are also solving quadratic equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Solving quadratic equations by factorising allows us to calculate values of the unknown variable in a quadratic equation using factorisation.
To do this we make sure the equation is equal to 0, factorise it into brackets and then solve the resulting linear equations.
E.g.
Solve the following quadratic equation by factorising:
So,
and
E.g.
Solve the following quadratic equation by factorising:
So,
and
In order to factor a quadratic algebraic equation we need to make sure it is in the form of the general quadratic equation:
We must ensure the quadratic equation is equal to 0, rearranging it if necessary.
NOTE: Quadratic equations are a type of polynomial equation because they consist of two or more algebraic terms.
First make sure that the equation is equal to 0.
Step 1: Fully factorise the quadratic.
Step 2: Set each bracket equal to 0.
Step 3: Solve each linear equation.
To recap factorising quadratics and solving linear equations check out of step by step lesson guides.
Step by step: Solving linear equations
Step by step: Factorising quadratics
Download for free one or more quadratic equations worksheets to help you practice this further and more.
DOWNLOAD FREEDownload for free one or more quadratic equations worksheets to help you practice this further and more.
DOWNLOAD FREEThis lesson is part of the larger topic, quadratic equations. It may be helpful to take a look at the quadratic equations page before moving on to the related lessons below:
See also: Maths equations
Solve
2We know that if two values multiply together to get 0, at least one of them must be 0. So set each bracket equal to 0.
3Solve each equation to find
The opposite of
The opposite of
When we solve a quadratic equation we normally have two solutions.
We call these the solutions, or roots, of the quadratic equation.
When we plot these values on an
We can see the real roots of the quadratic equation are where the quadratic graph crosses the
We can check that our solution is correct by substituting it into the original equation.
Solve
Fully factorise the quadratic expression.
We know that if two values multiply together to get 0, at least one of them must be 0. So put each bracket equal to 0.
Solve each equation to find x.
The opposite of
The opposite of +1 is -1, so -1 to both sides of the equation.
When we solve a quadratic equation we normally have two solutions.
We call these the solutions or roots of the quadratic equation.
We can see the real roots of the quadratic equation are where the quadratic graph crosses the
We can check that our solution is correct by substituting it into the quadratic function.
Do not try and square root the quadratic otherwise you will not get all the solutions!
When we multiply two values the order doesn’t matter.
E.g.
It is exactly the same here:
means
So,
is the same as
Don’t forget to set the factorised expression equal to zero and solve it.
Always check you have answered the question.
The term factorising can sometimes be written as ‘factoring’ or ‘factorization’.
If the quadratic equation involves two square terms (perfect squares) that are subtracted from each other, you will need to use the ‘difference of two squares’ to factorise it.
If a quadratic equation cannot be factorised, we can still solve it by using the quadratic formula.
To work out the number of real solutions a quadratic equation has we can use the discriminant. The derivation of the quadratic formula is fascinating, we will explore it more when we learn about ‘completing the square’.
It is possible to factorise a quadratic equation without ensuring that it is in the standard form of a quadratic equation, however this can be challenging.
1. Solve: {x}^2+5x+6=0
{x}^2+5x+6=0 can be factorised as (x+2)(x+3)=0 . Setting each bracket equal to zero and solving leads to the required solutions.
2. Solve: {x}^2-x-20=0
{x}^2-x-20=0 can be factorised as (x-5)(x+4)=0 . Setting each bracket equal to zero and solving leads to the required solutions.
3. Solve: 2{x}^2+3x-9=0
2{x}^2+3x-9=0 can be factorised as (2x-3)(x+3)=0 . Setting each bracket equal to zero and solving leads to the required solutions.
4. Solve: 3{x}^2-9x+6=0
3{x}^2-9x+6=0 can be factorised as 3(x-1)(x-2)=0 . Setting each bracket equal to zero and solving leads to the required solutions.
1. Factorise x^{2}-x-30
(2 marks)
(1)
(x+5) (x-6)
(1)
2. Hence, or otherwise, solve x^{2}-x-30=0
(1 mark)
x=-5 and x=6
(1)
3. Solve 2 x^{2}-5 x-3
(3 marks)
(1)
2x+1=0 and x-3=0
(1)
x=-\frac{1}{2} \text { and } x=3
(1)
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