Factorization of quadratic expressions

Factorization of Quadratic expression involves breaking down a mathematical expression of the form ax² + bx+ c, where a, b and c are constants and a≠0. a is called the coefficient of x², b, the coefficient of x and c the constant term.

To factorize a quadratic expression, we express it in such a way that the process of expansion can be worked backwards. This requires that the term in x be replaced by two terms chosen in such a way that the grouping method of factorization can be used.

Factorization of quadratic expressions is a key algebraic skill that helps us rewrite a quadratic expression into simpler, meaningful factors. Instead of working with an expanded form like $ax^2 + bx + c$ax2+bx+c, factorization allows us to express it as a product of two binomials, making it easier to solve equations, find roots, and understand the behavior of graphs. This topic builds on earlier ideas of factors and multiplication and plays an important role in many areas of mathematics, including graphing parabolas and solving real-life problems involving area, motion, and optimization.

factorization of quadratic expression when coefficient x2 = 2

consider the expression:

(i)(x-3)(x+4) = x²+4x-3x-12 = x²+ x-12

(ii)(x+5)(x+1)= x² +x+5x+5 = x² +6x+5

please note: the sum of constant terms in the brackets is equal to coefficients of x in the expanded term.

that is, -3 + 4 = 1 and 5+1=6

The coefficient of x in the first expression = 1 while the coefficient of x in the second expression =6.

The product of the constant factors in the products gives the value of the constant term in the expanded expression.

in the first expression, -3 x 4 = -12 and 1 x 5 = 5

conclusion:

1. The sum of the constant terms in the factors is equal to the coefficient of x in the expression
2. the product of the constant terms in the factors is equal to the constant terms of the expression

Example 1

Factorize the expression x²+6x+8

solution

the factors will be of the form (x + a)(x + b)

a+b = 6 and a x b = 8

we need to find the values of a and b that satisfies this conditions.

one of the best way is to list the products of 8. That is:

1 x 8 = 8

2 x 4 = 8

4 x 2 = 8

we add the coefficients of the products above:

1+8 = 9

2+4 = 6

so the factors of a and b are 2 and 4. Hence the factorized form of the expression is

(x+2)(x+4)

Revision Exercise

1. Factorize the following quadratic exprerssions

(a) x² – 5x +6 (b) x²+2x-35 (c)x²-10x+24

(d)x² +4x -21 (e) x² – 5x -6

Factorizing coefficient of x2 when a > 1

consider the expression: (4x+3)(2x+1)

we expand: 4x(2x+1)+3(2x+1) = 8x²+4x+6x+3 = 8x²+10x+3

From the quadratic expressions above, comparing ax²+bx+c

we see that a=8, b=10 and c=3

we cannot use product of 3 to get factors that make b which is 10.

but we should get product of ac, such that 3×8 = 24

product of 24 includes:

• 8×3
• 12×2
• 6×4

Among this, the product that gives the sum of 10 is 6 and 4

hence the factor bx = 6x+4x=10x

hence 8x²+10x+3 = 8x²+6x+4x+3

we then group them together into two: (8x²+6x)+(4x+3)

the expression then becomes : 2x(4x+3)+1(4x+3)

hence the factorized form of 8x²+10x+3 =(2x+1)(4x+3)

Generally, to factorize the expression of the form ax²+bx+c, find two numbers such that their product is ac and their sum is b. You should then split the middle term into two terms such that their coefficients are the numbers determined.

Example problem on factorization

Factorize 4x² +6x -18

solution

The product of coefficient of x² and that of c will be 4 x -18 = -72

The factors of -72 whose sum is 6 are -6 and 12 . That is; -6+12 = 6

splitting the bx term we get: 4x² +12x-6x-18 = 4x(x+3)-6(x+3)

=(4x-6)(x+3)

note that (4x-6) can be factorized father because 2 can divide both 4 and 6.

hence we have 2(2x-3)

hence 4x²+6x-18 = 2(2x-3)(x+3)

2. Factorize the following quadratic expressions:

(a) 2x² +3x -2 (b) 3x² – 2x -8 (c) 4x² +7x +3

(d) 5x² – 21x +4 (e) 3x² +11x +6 (f) 14x² – 16x +2

3. The square of a number is 5 more than three times the number. Find the number.
4. ABC is an isosceles triangle in which AB= AC. The size of angles ABC and ACB are (3x²-2x+4)^o and (9x-6)^o respectively. Calculate the two possible sets of values of the three angles of triangle ABC.
5. A picture measuring 6cm by 9cm is mounted on a frame so as to leave a uniform margin of width x cm all around. If the area of the margin is 36cm². find x

Related topics

• Introduction to matrices
• Quadratic Equations