Factorising trinomials: extension

Coefficient for x² greater than 1

When the coefficient for \({x^2}\) is greater than 1, there is a different method to follow.

Here is one method.

Remember \(a{x^2} + bx + c\)

Step 1:\(a \times c\) gives the number needed to find factors that will also add to give b.

Step 2: Once we have the correct factors, replace \(bx\) with these factors.

Step 3: Take a common factor of the first two terms then do the same for the last two terms.

Step 4: You should notice that you have two brackets the same. One bracket is for your answer and the other bracket contains the common factors in front of the brackets.

Example

Factorise \(3{x^2} - 7x - 6\)

Answer

\(a = 3,b = - 7,c = - 6\)

Step 1

\(a \times c = 3 \times - 6 = - 18\)

We need factors of -18 which add to give -7.

Factors of -18	Adding the factors
\(1x - 18\)	\(- 17\)
\(-1 x 18\)	\(17\)
\(2x - 9\)	\(- 7\)
\(-2 x 9\)	\(7\)
\(3x - 6\)	\(- 3\)
\(-3 x 6\)	\(3\)

Factors of -18	\(1x - 18\)
Adding the factors	\(- 17\)

Factors of -18	\(-1 x 18\)
Adding the factors	\(17\)

Factors of -18	\(2x - 9\)
Adding the factors	\(- 7\)

Factors of -18	\(-2 x 9\)
Adding the factors	\(7\)

Factors of -18	\(3x - 6\)
Adding the factors	\(- 3\)

Factors of -18	\(-3 x 6\)
Adding the factors	\(3\)

Required factors are 2 and -9.

Step 2

Replace \(7x\) with \(+2x\) and \(-9x\) in the order which helps to get common factors.

\(3x^{2}-9x+2x-6\)

Step 3

\(3x(x-3) +2(x-3)\)

(Take a common factor of the first two terms, then the last two terms)

Step 4

\((x - 3)(3x + 2)\)

Question

Factorise the following:

\(4{x^2} + 8x - 5\)

\(a = 4,b = 8,c = - 5\)

Step 1

\(a \times c = 4 \times - 5 = - 20\)

Factors of -20	Adding the factors
\(1x - 20\)	\(- 19\)
\(-1 x 20\)	\( 19\)
\(2x-10\)	\(-8\)
\(-2 x 10\)	\(8\)
\(4x -5\)	\(-1\)
\(-4x5\)	\(1\)

Factors of -20	\(1x - 20\)
Adding the factors	\(- 19\)

Factors of -20	\(-1 x 20\)
Adding the factors	\( 19\)

Factors of -20	\(2x-10\)
Adding the factors	\(-8\)

Factors of -20	\(-2 x 10\)
Adding the factors	\(8\)

Factors of -20	\(4x -5\)
Adding the factors	\(-1\)

Factors of -20	\(-4x5\)
Adding the factors	\(1\)

Required factors are -2 and 10.

Step 2

Replace +8x with -2x and +10x in the order which helps to get common factors.

\(4{x^2} - 2x + 10x - 5\)

(Replace)

Step 3

\(2x(2x - 1) + 5(2x - 1)\)

(Common factors)

Step 4

\((2x - 1)(2x + 5)\)

Question

Factorise \(10{x^2} + 38x - 8\)

Next up

Test your understanding

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Factorising an algebraic expressionFactorising trinomials: extension