Algebra Know-It-ALL

502 Review Questions and Answers

where a and b are coefficients of the variable x, and c is a constant. For the equation to be a

quadratic, the coefficient of x^2 (in this case a) must not be equal to 0.

Question 22-2

How can we write the following equation as a quadratic in polynomial standard form when

a 1 ,a 2 ,b 1 , and b 2 are real numbers, and x is the variable?

(a 1 x+b 1 )(a 2 x+b 2 )= 0

Answer 22-2

We can start by multiplying the two binomials on the left side together, using the product of

sums rule from Chap. 9. When we do that, we get

a 1 a 2 x^2 +a 1 b 2 x+b 1 a 2 x+b 1 b 2 = 0

which can be rearranged to obtain

(a 1 a 2 )x^2 + (a 1 b 2 +b 1 a 2 )x+b 1 b 2 = 0

This equation is in polynomial standard form, provided a 1 ≠ 0 and a 2 ≠ 0. The coefficient a

from the “template” (Answer 22-1) is the quantity a 1 a 2. The coefficient b from the “template”

is the quantity (a 1 b 2 +b 1 a 2 ). The constant c from the “template” is the quantity b 1 b 2.

Question 22-3

Which of the following equations are quadratics in one variable? Which are not?

x^2 = 8 x+ 3 x^3

− 3 x= 7 x^2 − 12

x^2 + 2 x= 7 −x

x^4 − 2 =− 8 x^2 − 7 x^3

13 + 3 x= 12 x^2

Answer 22-3

A quadratic equation in one variable always contains a nonzero multiple of the variable squared,

and no higher powers of the variable. There may also be terms containing the variable itself (to

the first power) along with terms that are simple constants. On that basis, the second, third,

and fifth equations are quadratics in one variable. The first and fourth equations are not.

Question 22-4

What are the roots of the following quadratic equation, where a 1 ,a 2 ,b 1 , and b 2 are real num-

bers, and x is the variable? Assume that neither a 1 nor b 1 is equal to 0:

(a 1 x+b 1 )(a 2 x+b 2 )= 0