Algebra 1 Common Core Student Edition, Grade 8-9

Using the Discriminant

How many real-number solutions does 2 x 2 — 3x = - 5 have?

Pl an

Can you solve this

problem another

way?

Yes. You could a ctu a lly

solve the equation to find

any solutions. However,

you only need to know

the number o f solutions,

so use th e discrim inant.

Write the equation in

standard form.

Evaluate the

discriminant by

substituting 2 fo r a,

-3 for b, and 5 fore.

Draw a conclusion.

2x 2 - 3x + 5 = 0

b 2 - 4ac = (-3)2 - 4(2)(5)

= -3 1

B e c au se t h e d is c rim in a n t is

negative, the equation has

no r e a l- n u m b e r s o lu tio n s.

& Got It? 4. a. How many real-number solutions does 6x2 - 5x = 7 have?

b. Reasoning If a is positive and c is negative, how many real-number

solutions will the equation ax2 + bx + c = 0 have? Explain.

Lesso n Ch eck

Do y o u k n o w H OW?

Use the quadratic formula to solve each equation. If

necessary, round answers to the nearest hundredth.

1. —3x2 — llx + 4 = 0


2. 7x2 - 2x = 8


3. How many real-number solutions does the equation
   -2 x 2 + 8x - 5 = 0 have?

1

MATHEMATICAL

Do y o u UN DERSTAND? y& J PRACTICES

@ 4. Vocabulary Explain how the discriminant of the

equation ax2 + bx + c = 0 is related to the number

of x-intercepts of the graph of y = ax2 + bx + c.

5. Reasoning What method would you use to solve the
   equation x2 + 9x + c = 0 if c = 14? If c = 7? Explain.
   @ 6. W r i t i n g Explain how completing the square is used
   to derive the quadratic formula.

Pr act i ce an d Pr o b l em - So l v i n g Ex er ci ses (g j pr a ct ices

^jjj| Practice Use the quadratic formula to solve each equation. ^ See Problem 1.

7. 2x2 + 5x + 3 = 0 8. 5x2 + 16x - 84 = 0 9. 4x2 + 7x - 15 = 0


8. 3x2-4 1 x = -110 11. 18x2 - 4 5 x - 5 0 = 0 12. 3x2 + 44x = -9 6


9. 3x2 + 19x= 154 14. 2x2 - x - 120 = 0 15. 5x2 - 47x= 156

586 Ch ap t er 9 Qu ad r at i c Fu n ct i o n s an d Eq u at i o n s