Algebra Tutorials!  
     
     
Tuesday 21st of November
   
Home
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
Mixed
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
Polynomials
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Numbers
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
Polynomials
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Exponents
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
   
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Quadratic Expressions

In the package on Factorizing Expressions we looked at how to factorize quadratic expressions which have the number 1 in front of the highest order term, x, y, z, etc.. If the highest order term has a number other than this then more work must be done to factorize the expression. As in the earlier case, some insight is gained by looking at a general expression with factors (ax + c) and (bx + d). Then

showing that the coeffcient of the square term, x, is ab, the product of the coeffcients of the x-terms in each factor. The coeffcient of the x-term is made up from the coeffcients as follows:

This is the information needed to find the factors of quadratic expressions.

Example 1

Factorize the following expressions.

(a) 2x + 7x + 3 , (b) 10x + 9x + 2 .

Solution

(a) The factors of 2 are 2 and 1, and the factors of 3 are 3 and 1. If the quadratic expression factorizes then it is likely to be of the form (2x + c)(1x + d) and the choice for c, d is 3, 1 or 1, 3. Trying the first combination,

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

= 2x + 5x + 3 (which is incorrect) .

The second choice is

(2x + 1)(x + 3) = 2x+ 6x + x + 3 ,

= 2x+ 7x + 3 , which is therefore the correct factorization.

(b) There is more than one choice for the first term since 10 is 1 × 10 as well as 2 × 5. The final term will factor as 2 × 1. Which combination of pairs, either (1, 10) with (2, 1), or (2, 5) with (2, 1), will give the correct coeffcient of x, i.e., 9? The latter two pairs seem the more likely since 2 × 2 + 5 × 1 = 9. Checking

(2x + 1)(5x + 2) = 10x + 4x + 5x + 2 ,

= 10x + 9x + 2 .

Exercise 1.

Factorize each of the following expressions.

(a) 2x + 5x + 3

(b) 3x + 7x + 2

(c) 3y - 5y - 2

(d) 4z - 23z + 15

(e) 64z + 4z - 3

(f) 4w - 25

Solution

(a) In this case we have 2x + 5x + 3 = (2x + 3)(x + 1)

(b) In this case we have 3x + 7x + 2 = (3x + 1)(x + 2)

(c) In this case we have 3y - 5y - 2 = (3y + 1)(y - 2)

(d) In this case we have 4z - 23z + 15 = (4z - 3)(z - 5)

(e) In this case we have 64z + 4z - 3 = (16z - 3)(4z + 1)

(f) This is a case of the difference of two squares which was seen in the package on Brackets. 4w - 25 = (2w - 5)(2w + 5)

Quiz

To which of the following does 12x2 + 17x - 14 factorize?

(a) (12x + 7)(x - 2) (b) (x + 2)(12x - 7) (c) (4x + 7)(x - 3) (d) (x - 7)(4x + 3)

Solution

There are several possibilities since the final term is -14 and the two quantities corresponding to c and d must therefore have opposite signs. The possible factors of 12 are (1, 12), (2, 6), (3, 4). For -14, the possible factors are (±1,14), (±2,7). It is now a matter of trial and error. The possible combinations are

(1, 12) and (±1,14) , (1, 12) and (±2,7) ,

(2, 6) and (±1,14) , (2, 6) and (±2,7) ,

(3, 4) and (±1,14) , (3, 4) and (±2,7) .

By inspection (2 × 12) + (1 × {-7}) = 24 - 7 = 17, so the factors appear to be (x + 2) and (12x - 7). This can easily be checked.

(x + 2)(12x - 7) = 12x - 7x + 24x - 14 ,

= 12x+ 17x - 14 ,

and the required factorization has been achieved.

 
Copyrights © 2005-2017