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Determine the number of solutions to each quadratic equation:

8 m 2 3 m + 6 = 0 5 z 2 + 6 z 2 = 0 9 w 2 + 24 w + 16 = 0 9 u 2 2 u + 4 = 0

no real solutions 2 1 no real solutions

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Determine the number of solutions to each quadratic equation:

b 2 + 7 b 13 = 0 5 a 2 6 a + 10 = 0 4 r 2 20 r + 25 = 0 7 t 2 11 t + 3 = 0

2 no real solutions 1 2

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Identify the most appropriate method to use to solve a quadratic equation

We have used four methods to solve quadratic equations:

  • Factoring
  • Square Root Property
  • Completing the Square
  • Quadratic Formula

You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.

Identify the most appropriate method to solve a quadratic equation.

  1. Try Factoring first. If the quadratic factors easily, this method is very quick.
  2. Try the Square Root Property next. If the equation fits the form a x 2 = k or a ( x h ) 2 = k , it can easily be solved by using the Square Root Property.
  3. Use the Quadratic Formula . Any quadratic equation can be solved by using the Quadratic Formula.

What about the method of completing the square? Most people find that method cumbersome and prefer not to use it. We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.

Identify the most appropriate method to use to solve each quadratic equation:

5 z 2 = 17 4 x 2 12 x + 9 = 0 8 u 2 + 6 u = 11

Solution

5 z 2 = 17

Since the equation is in the a x 2 = k , the most appropriate method is to use the Square Root Property.


4 x 2 12 x + 9 = 0

We recognize that the left side of the equation is a perfect square trinomial, and so Factoring will be the most appropriate method.


8 u 2 + 6 u = 11

Put the equation in standard form. 8 u 2 + 6 u 11 = 0

While our first thought may be to try Factoring, thinking about all the possibilities for trial and error leads us to choose the Quadratic Formula as the most appropriate method

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Identify the most appropriate method to use to solve each quadratic equation:

x 2 + 6 x + 8 = 0 ( n 3 ) 2 = 16 5 p 2 6 p = 9

factor Square Root Property Quadratic Formula

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Identify the most appropriate method to use to solve each quadratic equation:

8 a 2 + 3 a 9 = 0 4 b 2 + 4 b + 1 = 0 5 c 2 = 125

Quadratic Formula factoring Square Root Property

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Key concepts

  • Quadratic Formula The solutions to a quadratic equation of the form a x 2 + b x + c = 0 , a 0 are given by the formula:
    x = b ± b 2 4 a c 2 a
  • Solve a Quadratic Equation Using the Quadratic Formula
    To solve a quadratic equation using the Quadratic Formula.
    1. Write the quadratic formula in standard form. Identify the a , b , c values.
    2. Write the quadratic formula. Then substitute in the values of a , b , c .
    3. Simplify.
    4. Check the solutions.
  • Using the Discriminant, b 2 4 a c , to Determine the Number of Solutions of a Quadratic Equation
    For a quadratic equation of the form a x 2 + b x + c = 0 , a 0 ,
    • if b 2 4 a c > 0 , the equation has 2 solutions.
    • if b 2 4 a c = 0 , the equation has 1 solution.
    • if b 2 4 a c < 0 , the equation has no real solutions.
  • To identify the most appropriate method to solve a quadratic equation:
    1. Try Factoring first. If the quadratic factors easily this method is very quick.
    2. Try the Square Root Property next. If the equation fits the form a x 2 = k or a ( x h ) 2 = k , it can easily be solved by using the Square Root Property.
    3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Practice makes perfect

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

4 m 2 + m 3 = 0

m = −1 , m = 3 4

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2 p 2 7 p + 3 = 0

p = 1 2 , p = 3

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p 2 + 7 p + 12 = 0

p = −4 , p = −3

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r 2 8 r 33 = 0

r = −3 , r = 11

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3 u 2 + 7 u 2 = 0

u = −7 ± 73 6

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2 a 2 6 a + 3 = 0

a = 3 ± 3 2

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2 x 2 + 3 x + 9 = 0

no real solution

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v ( v + 5 ) 10 = 0

v = −5 ± 65 2

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3 w ( w 2 ) 8 = 0

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1 3 m 2 + 1 12 m = 1 4

m = −1 , m = 3 4

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16 c 2 + 24 c + 9 = 0

c = 3 4

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25 d 2 60 d + 36 = 0

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5 m 2 + 2 m 7 = 0

m = 7 5 , m = 1

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p 2 6 p 27 = 0

p = −3 , p = 9

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4 r 2 + 3 r 5 = 0

r = −3 ± 89 8

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2 a 2 + 12 a + 5 = 0

a = −6 ± 26 2

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3 4 b 2 + 1 2 b = 3 8

b = −2 ± 11 6

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2 x 2 + 12 x 3 = 0

x = −6 ± 42 4

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Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions to each quadratic equation.


4 x 2 5 x + 16 = 0
36 y 2 + 36 y + 9 = 0
6 m 2 + 3 m 5 = 0
18 n 2 7 n + 3 = 0

no real solutions 1
2 no real solutions

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9 v 2 15 v + 25 = 0
100 w 2 + 60 w + 9 = 0
5 c 2 + 7 c 10 = 0
15 d 2 4 d + 8 = 0

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r 2 + 12 r + 36 = 0
8 t 2 11 t + 5 = 0
4 u 2 12 u + 9 = 0
3 v 2 5 v 1 = 0

1 no real solutions
1 2

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25 p 2 + 10 p + 1 = 0
7 q 2 3 q 6 = 0
7 y 2 + 2 y + 8 = 0
25 z 2 60 z + 36 = 0

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Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.


x 2 5 x 24 = 0
( y + 5 ) 2 = 12
14 m 2 + 3 m = 11

factor square root
Quadratic Formula

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( 8 v + 3 ) 2 = 81
w 2 9 w 22 = 0
4 n 2 10 = 6

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6 a 2 + 14 = 20
( x 1 4 ) 2 = 5 16
y 2 2 y = 8

factor square root
factor

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8 b 2 + 15 b = 4
5 9 v 2 2 3 v = 1
( w + 4 3 ) 2 = 2 9

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Everyday math

A flare is fired straight up from a ship at sea. Solve the equation 16 ( t 2 13 t + 40 ) = 0 for t , the number of seconds it will take for the flare to be at an altitude of 640 feet.

5 seconds, 8 seconds

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An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation 1 2 h 2 + 3 h = 140 for h , the height of the window.

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Writing exercises

Solve the equation x 2 + 10 x = 200
by completing the square
using the Quadratic Formula
Which method do you prefer? Why?

−20 , 10 −20 , 10
answers will vary

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Solve the equation 12 y 2 + 23 y = 24
by completing the square
using the Quadratic Formula
Which method do you prefer? Why?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four rows and four columns. The first row is a header row and it labels each column. The first column is labeled

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
KAJAL Reply
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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