Intermediate Algebra (11th edition)

Calculating the discriminant can also help you decide how to solve a quadratic

equation. If the discriminant is a perfect square ( including 0), then the equation

can be solved by factoring. Otherwise, the quadratic formula should be used.

Using the Discriminant

Find the discriminant. Use it to predict the number and type of solutions for each

equation. Tell whether the equation can be solved by factoring or whether the quad-

ratic formula should be used.

(a)

We find the discriminant by evaluating Because the value

of bin this equation is

Apply the exponent. Multiply.

= 361 , or 192 , which is a perfect square.

= 1 + 360

= 1 - 122 - 41621 - 152 a=6, b=-1, c=- 15

b^2 - 4 ac

- 1.

b^2 - 4 ac. -x= - 1 x,

6 x^2 - x- 15 = 0

EXAMPLE 4

SECTION 9.2 The Quadratic Formula 509

so

b= 1.

x= 1 x,

Because the discriminant is negative and a, b, and care integers, this equation will

have two nonreal complex solutions. The quadratic formula should be used to

solve it.

(d) Write in standard form as

Apply the exponent. Multiply.

Subtract.

The discriminant is 0, so the quantity under the radical in the quadratic formula is 0,

and there is only one rational solution. The equation can be solved by factoring.

NOW TRY

= 0

= 144 - 144

= 1 - 1222 - 4142192 a=4, b=-12, c= 9

b^2 - 4 ac

4 x^2 + 9 = 12 x 4 x^2 - 12 x+ 9 = 0.

Use parentheses and

substitute carefully.

Since a, b, and care integers and the discriminant 361 is a perfect square, there will

be two rational solutions. The equation can be solved by factoring.

(b) Write in standard form as

Apply the exponent. Multiply.

Add.

Because 76 is positive but notthe square of an integer and a, b, and care integers,

the equation will have two irrational solutions and is best solved using the quadratic

formula.

(c)

Apply the exponent. Multiply.

= - 15 Subtract.

= 1 - 16

= 12 - 4142112 a=4, b=1, c= 1

b^2 - 4 ac

4 x^2 +x+ 1 = 0

= 76

= 16 + 60

= 1 - 422 - 41321 - 52 a=3, b=-4, c=- 5

b^2 - 4 ac

3 x^2 - 4 x= 5 3 x^2 - 4 x- 5 = 0.

NOW TRY
EXERCISE 4
Find each discriminant. Use it
to predict the number and type
of solutions for each equation.
Tell whether the equation can
be solved by factoring or
whether the quadratic formula
should be used.

(a)

(b)

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

9 x^2 = 24 x- 16

8 x^2 - 6 x- 5 = 0

NOW TRY ANSWERS

4. (a)196; two rational solutions;
   factoring
   (b)0; one rational solution;
   factoring
   (c) ; two nonreal complex
   solutions; quadratic formula

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