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4-6 Study Guide and InterventionThe Quadratic Formula and the Discriminant 

Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form

ax2+bx+c  = 0.

Quadratic Formula The solutions of ax2+bx+c  = 0, with a ≠ 0, are given ! x =

−b±√ b2−4ac

2a"

Example: Solve  x2−5 x  = 14 by using the Quadratic Formula.

Rewrite the equation as  x2  – 5 x – 14 = 0.

 x =−b±√ b

2−4ac2a

#ua$rati% &or'ula

=−(−5)±√ (−5)

2−4(1)(−14)

2(1)

Re(la%e a with ), b with *+, an$ c with *)"

=5±√ 81

2-i'(lif!"

=5±9

2

= 7 or –2

The solutions are –2 and 7.

Exercises

Solve each equation by using the Quadratic Formula.

1.  x2   2 x – !5 = 0 .  x

2   10 x 24 = 0 !.  x2  – 11 x 24 = 0

4. 4   x2   1" x – 5 = 0 ". 14   x

2   " x 1 = 0 #. 2   x2  – x – 15 = 0

$. !   x2   5 x = 2 %. 2 y

2   y – 15 = 0 &. !   x2  – 1# x 1# = 0

1'. $   x2   # x – " = 0 11. r

2  –3 r

5 

2

25 = 0 1.  x

2  – 10 x – 50 =

0

1!.  x2   # x – 2! = 0 14. 4   x

2  – 12 x – #! = 0 1".  x2  – # x 21 = 0

.ha(ter   36 Glencoe Algebra

8/16/2019 8 - The Quadratic Formula and the Discriminant

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.ha(ter 37 Glencoe Alg

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4-6 Study Guide and Intervention (continued)The Quadratic Formula and the Discriminant 

(oots and the )iscriminant

DiscriminantThe e/(ression un$er the ra$i%al sign, b

2  * ac , in the #ua$rati% &or'ula is %alle$ th

discriminant"

Discriminant Type and Number of Roots

b2  * ac 0 an$ a (erfe%t s1uare 2 rational roots

b2

 * ac 0, ut not a (erfe%t s1uare 2 irrational roots

b2  * ac = 0 ) rational root

b2

 * ac 3 0 2 %o'(le/ roots

Example: Find the value o* the discriminant *or each equation. +hen describe the number and type o*roots *or the equation.

a.   x2  , " x , !

The discriminant is b2  – 4ac = 5

2  – 4%2&

%!& or 1. The discriminant is a 'erfect square( sothe equation has 2 rational roots.

b. !   x2   –  x , "

The discriminant is b2  – 4ac = (−2)2  –

4%!& %5& or

 –5#. The discriminant is ne)ative( so the equation

has 2 com'le* roots.

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