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By McGraw-Hill Professional
Updated on Sep 27, 2011

## Introduction to Quadratic Type Expressions

An expression with three terms where the power of the first term is twice that of the second and the third term is a constant is called a quadratic type expression. They factor in the same way as quadratic polynomials. The power on x in the factorization will be the power on x in the middle term. To see the effect of changing the exponents, let us look at x 2 – 2 x – 3 = ( x – 3)( x + 1).

x 4 – 2 x 2 – 3 = ( x 2 – 3)( x 2 + 1)

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

x 10 – 2 x 5 – 3 = ( x 5 – 3)( x 5 + 1)

x –4 – 2 x –2 – 3 = ( x –2 – 3)( x –2 + 1)

x 2/3 – 2 x 1/3 – 3 = ( x 1/3 – 3)( x 1/3 + 1)

x 1 – 2 x 1/2 – 3 = ( x 1/2 – 3)( x 1/2 + 1)

## Quadratic Type Expressions Practice Problems

#### Practice

1. x 4 – 3 x 2 + 2 =

2. x 10 – 3 x 5 + 2 =

3. x 2/5 – 3 x 1/5 + 2 =

4. x –6 – 3 x –3 + 2 =

5. x 1/2 – 3 x 1/4 + 2 =

6. x 4 + 10 x 2 + 9 =

7. x 6 – 4 x 3 – 21 =

8. 4 x 6 + 4 x 3 – 35 =

9. 10 x 10 + 23 x 5 + 6 =

10. 9 x 4 – 6 x 2 + 1 =

11. x 2/7 – 3 x 1/7 – 18 =

12. 6 x 2/3 – 7 x 1/3 – 3 =

13. x 1/3 + 11 x 1/6 + 10 =

14. 15 x 1/2 – 8 x 1/4 + 1 =

16. 25 x 6 + 20 x 3 + 4 =

#### Solutions

1. x 4 – 3 x 2 + 2 = ( x 2 – 2)( x 2 – 1)

2. x 10 – 3 x 5 + 2 = ( x 5 – 2)( x 5 – 1)

3. x 2/5 – 3 x 1/5 + 2 = ( x 1/5 – 2)( x 1/5 – 1)

4. x –6 – 3 x –3 + 2 = ( x –3 – 2)( x –3 – 1)

5. x 1/2 – 3 x 1/4 + 2 = ( x 1/4 – 2)( x 1/4 –1)

6. x 4 + 10 x 2 + 9 = ( x 2 + 9)( x 2 + 1)

7. x 6 – 4 x 3 – 21 = ( x 3 – 7)( x 3 + 3)

8. 4 x 6 + 4 x 3 – 35 = (2 x 3 + 7)(2 x 3 – 5)

9. 10 x 10 + 23 x 5 + 6 = (10 x 5 + 3)( x 5 + 2)

10. 9 x 4 – 6 x 2 + 1 = (3 x 2 – 1)(3 x 2 – 1) = (3 x 2 – 1) 2

11. x 2/7 – 3 x 1/7 – 18 = ( x 1/7 – 6)( x 1/7 + 3)

12. 6 x 2/3 – 7 x 1/3 – 3 = (2 x 1/3 – 3)(3 x 1/3 + 1)

13. x 1/3 + 11 x 1/6 + 10 = ( x 1/6 + 10)( x 1/6 + 1)

14. 15 x 1/2 – 8 x 1/4 + 1 = (3 x 1/4 – 1)(5 x 1/4 – 1)

16. 25 x 6 + 20 x 3 + 4 = (5 x 3 + 2)(5 x 3 + 2) = (5 x 3 + 2) 2

Practice problems for this concept can be found at: Algebra Factoring Practice Test.

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