Algebra Know-It-ALL

504 Review Questions and Answers

where a and b are coefficients of the variable x,a≠ 0, and c is a constant. What is the general

formula for solving this quadratic?

Answer 22-7

The formula, known as the quadratic formula, is

x= [−b± (b^2 − 4 ac)1/2] / (2a)

This is worth memorizing!

Question 22-8

What is the discriminant in the quadratic formula? Why is it significant?

Answer 22-8

The discriminant, sometimes symbolized as d, is the quantity (b^2 − 4 ac). It tells us whether or

not a quadratic equation with real-number coefficients and a real-number constant has any

real roots. If d > 0, then the equation has two different real roots. If d= 0, then the equation

has one real root with multiplicity 2. If d < 0, then the equation has no real roots.

Question 22-9

How can we use the quadratic formula to find the roots of the following equation?

9 x^2 − 42 x=− 49

Answer 22-9

Before applying the formula, we must get the equation into polynomial standard form. We

can do that by adding 49 to each side, obtaining

9 x^2 − 42 x+ 49 = 0

In the general polynomial standard equation

ax^2 +bx+c= 0

we have a= 9, b=−42, and c= 49. Plugging these into the quadratic formula, we get

x= [−b± (b^2 − 4 ac)1/2] / (2a)

= [42 ± (42^2 − 4 × 9 × 49)1/2] / (2 × 9)

= [42 ± (1,764 − 1,764)1/2] / 18

= 42/18

= 7/3

This equation has the single root x= 7/3 with multiplicity 2.