A quadratic in the form ax2+bx+c can be factorised. You have already learnt how to factorise a quadratic when a=1 by finding two numbers to multiply to give c and add to give b. This lesson will also discuss how to factorise a quadratic expression where a>1.
Factorise when a=1
A quadratic is in the form
ax2+bx+c
When a=1 we can factorise a quadratic as follows:
PROCEDURE
1.
Ensure the quadratic is in the form ax2+bx+c and identifya,b andc
2.
Set up the answer with two brackets and an x at the start of each bracket, as follows(x)(x)
3.
Find two numbers which multiply to give c and add to giveb
4.
Fill these numbers in the double brackets
Example 1
Factorise
x2+10x+24
The quadratic is in the correct form whereb=10andc=24.
6×4=24and6+4=10,so the numbers are6and4.
x2+10x+24=(x+6)(x+4)
Factorise when a>1
When a>1 you can factorise a quadratic as follows:
PROCEDURE
1.
Ensure the quadratic is in the form ax2+bx+c, and identify a,b and c.
2.
Multiply a and c to give a new number d.
3.
Think of two numbers, m and n which multiply to give d and add to give b.
4.
Write the quadratic using m and n in the form ax2+mx+nx+c.
5.
Factorise ax2+mx into single brackets.
6.
Factorise nx+c into single brackets.
7.
Use both factorised single brackets expressions to find the double brackets expression. The first bracket consists of the terms outside the single brackets, and the second bracket consists of the expression inside the single brackets.
Example 2
Factorise
2x2+13x+15
The quadratic hasa=2,b=13andc=15. Multiplyaandc to give2×15=30.
Think of two numbers which will multiply to give30 and add to give13. The two numbers are10and3, as10×3=30and10+3=13.
Write the quadratic expression using10 and3, and factorise.
The two expressions inside both single brackets should match, in this casex+5.
The final answer consists of two brackets, the first bracket contains the terms on the outside of each bracket,2xand+3. And the second bracket is the repeated expression inside the single brackets factorised.
(2x+3)(x+5)
Example 3
Factorise and solve
3x2+11x−4=0
The quadratic has a=3,b=11andc=−4. Multiplyaandcto give−12.
Think of two numbers which will multiply to give−12 and add to give+11. The two numbers are+12and−1, as 12×−1=−12 and 12−1=11.
Write the quadratic expression using 12 and −1, and factorise.
Now the quadratic is factorised, it can be solved. Find the solutions by making either bracket equal to 0.
x=31orx=−4
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FAQs - Frequently Asked Questions
How do you factorise a quadratic ax^2+bx+c when a>1?
When a quadratic has a>1, the method of factorising involves splitting b up to make two separate terms in x. The modified expression can then be separated into two separate groups and each group can be factorised using single brackets. The two single bracket expressions can then be used to factorise the quadratic into double brackets.
How do you factorise into double brackets?
Put the quadratic into the form ax^2 + bx + c. If a=1, then factorise the quadratic by finding two numbers that multiply to give c, and add to give b. These two numbers will go into the double brackets. e.g. x^2 + 7x + 10 = (x+2)(x+5).
What do double brackets mean?
Double brackets consist of two binomial expressions multiplied together, e.g. (x+1)(y-3). Each term in the first bracket should be multiplied by each term in the second bracket.