FACTORING QUADRATIC EXPRESSIONS BY SPLITTING THE MIDDLE TERMS

A quadratic polynomial is of the form of

ax2 + bx + c = 0

where a, b, c are real numbers.

How to factor quadratic expressions ?

Step 1 :

Write the product of the coefficient of the first and last term with their signs and the product of the factors equal to that particular value and when we simplify those two we should get the middle term.

Step 2 :

Replace the middle term by the sum of factors.

Step 3 :

Using grouping method, find the factors.

Problem 1 :

2x2 – 9x - 5

Solution :

Step 1 :

Product of factors = -10

Sum of factors = -9

Step 2 :

(-5) · 2 = -10 and (-5) + 2 = -3 ≠ -9

1 · (-10) = -10 and 1 + (-10) = -9

So, the factors are -10 and 1.

Step 3 :

= 2x2 + 1x – 10x – 5

By grouping,

= (2x2 + 1x) + (-10x – 5)

Step 4 :

By taking the common factor, we get

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

= (x – 5)(2x + 1)

Problem 2 :

3x2 + 5x - 2

Solution :

Step 1 :

Product of factors = -6

Sum of factors = 5

Step 2 :

(-2) · 3 = -6 and (-2) + 3 = 1 ≠ 5

(-1) · 6 = -6 and (-1) + 6 = 5

So, the factors are -1 and 6.

Step 3 :

= 3x2 - 1x + 6x – 2

By grouping,

= (3x2 - 1x) + (6x – 2)

Step 4 :

By taking the common factor, we get

= x(3x – 1) + 2(3x - 1)

= (x + 2)(3x – 1)

Problem 3 :

3x2 – 5x - 2

Solution :

Step 1 :

Product of factors = -6

Sum of factors = -5

Step 2 :

(-2) · 3 = -6 and (-2) + 3 = 1 ≠ -5

1 · (-6) = -6 and 1 + (-6) = -5

So, the factors are 1 and -6.

Step 3 :

= 3x2 + 1x - 6x – 2

By grouping,

= (3x2 + 1x) + (-6x - 2)

Step 4 :

By taking the common factor, we get

= x(3x + 1) - 2(3x + 1)

= (x - 2)(3x + 1)

Problem 4 :

2x2 + 3x - 2

Solution :

Step 1 :

Product of factors = -4

Sum of factors = 3

Step 2 :

Finding the factors of product and sum.

(-2) · 2 = -4 and (-2) + 2 = 0 ≠ 3

(-1) · 4 = -4 and (-1) + 4 = 3

So, the factors are -1 and 4.

Step 3 :

= 2x2 - 1x + 4x – 2

By grouping,

= (2x2 - 1x) + (4x – 2)

Step 4 :

By taking the common factor, we get

= x(2x – 1) + 2(2x - 1)

= (x + 2)(2x – 1)

Problem 5 :

2x2 + 3x - 5

Solution :

Step 1 :

Product of factors = -10

Sum of factors = 3

Step 2 :

(-1) · 10 = -10 and (-1) + 10 = 9 ≠ 3

(-2) · 5 = -10 and (-2) + 5 = 3

So, the factors are -2 and 5.

Step 3 :

= 2x2 - 2x + 5x – 5

By grouping,

= (2x2 - 2x) + (5x – 5)

Step 4 :

By taking the common factor, we get

= 2x(x – 1) + 5(x - 1)

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

Problem 6 :

5x2 – 14x - 3

Solution :

Step 1 :

Product of factors = -15

Sum of factors = -14

Step 2 :

(-3) · 5 = -15 and (-3) + 5 = 2 ≠ -14

1 · (-15) = -15 and 1 + (-15) = -14

So, the factors are 1 and -15.

Step 3 :

= 5x2 + x - 15x - 3

By grouping,

= (5x2 + x) + (-15x - 3)

Step 4 :

By taking the common factor, we get

= x(5x + 1) - 3(5x + 1)

= (x - 3)(5x + 1) Apart from the stuff given above if you need any other stuff in math, please use our google custom search here.

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