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Solving a quadratic equation with the quadratic formula

Use the quadratic formula to solve x 2 + x + 2 = 0.

First, we identify the coefficients: a = 1 , b = 1 , and c = 2.

Substitute these values into the quadratic formula.

x = b ± b 2 4 a c 2 a = ( 1 ) ± ( 1 ) 2 ( 4 ) ( 1 ) ( 2 ) 2 1 = 1 ± 1 8 2 = 1 ± 7 2 = 1 ± i 7 2

The solutions to the equation are 1 + i 7 2 and 1 i 7 2 or 1 2 + i 7 2 and 1 2 i 7 2 .

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Solve the quadratic equation using the quadratic formula: 9 x 2 + 3 x 2 = 0.

x = 2 3 , x = 1 3

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The discriminant

The quadratic formula    not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant    , or the expression under the radical, b 2 4 a c . The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. [link] relates the value of the discriminant to the solutions of a quadratic equation.

Value of Discriminant Results
b 2 4 a c = 0 One rational solution (double solution)
b 2 4 a c > 0 , perfect square Two rational solutions
b 2 4 a c > 0 , not a perfect square Two irrational solutions
b 2 4 a c < 0 Two complex solutions

The discriminant

For a x 2 + b x + c = 0 , where a , b , and c are rational and real numbers, the discriminant    is the expression under the radical in the quadratic formula: b 2 4 a c . It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Using the discriminant to find the nature of the solutions to a quadratic equation

Use the discriminant to find the nature of the solutions to the following quadratic equations:

  1. x 2 + 4 x + 4 = 0
  2. 8 x 2 + 14 x + 3 = 0
  3. 3 x 2 5 x 2 = 0
  4. 3 x 2 10 x + 15 = 0

Calculate the discriminant b 2 4 a c for each equation and state the expected type of solutions.

  1. x 2 + 4 x + 4 = 0

    b 2 4 a c = ( 4 ) 2 4 ( 1 ) ( 4 ) = 0. There will be one rational double solution.

  2. 8 x 2 + 14 x + 3 = 0

    b 2 4 a c = ( 14 ) 2 4 ( 8 ) ( 3 ) = 100. As 100 is a perfect square, there will be two rational solutions.

  3. 3 x 2 5 x 2 = 0

    b 2 4 a c = ( −5 ) 2 4 ( 3 ) ( −2 ) = 49. As 49 is a perfect square, there will be two rational solutions.

  4. 3 x 2 −10 x + 15 = 0

    b 2 4 a c = ( −10 ) 2 4 ( 3 ) ( 15 ) = −80. There will be two complex solutions.

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Using the pythagorean theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem    . It is based on a right triangle, and states the relationship among the lengths of the sides as a 2 + b 2 = c 2 , where a and b refer to the legs of a right triangle adjacent to the 90° angle, and c refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.

We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.

The Pythagorean Theorem is given as

a 2 + b 2 = c 2

where a and b refer to the legs of a right triangle adjacent to the 90 angle, and c refers to the hypotenuse, as shown in [link] .

Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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