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Key equations

quadratic formula x = b ± b 2 4 a c 2 a

Key concepts

  • Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See [link] , [link] , and [link] .
  • Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See [link] and [link] .
  • Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See [link] and [link] .
  • Completing the square is a method of solving quadratic equations when the equation cannot be factored. See [link] .
  • A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See [link] .
  • The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See [link] .
  • The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See [link] .

Section exercises

Verbal

How do we recognize when an equation is quadratic?

It is a second-degree equation (the highest variable exponent is 2).

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When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form a x 2 + b x + c = 0 we may graph the equation y = a x 2 + b x + c and have no zeroes ( x -intercepts).

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When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

We want to take advantage of the zero property of multiplication in the fact that if a b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0   o r  b = 0.

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In the quadratic formula, what is the name of the expression under the radical sign b 2 4 a c , and how does it determine the number of and nature of our solutions?

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Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.

One, when no linear term is present (no x term), such as x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d .

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Algebraic

For the following exercises, solve the quadratic equation by factoring.

x 2 9 x + 18 = 0

x = 6 , x = 3

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6 x 2 + 17 x + 5 = 0

x = 5 2 , x = 1 3

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3 x 2 75 = 0

x = 5 , x = −5

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4 x 2 = 9

x = 3 2 , x = 3 2

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5 x 2 = 5 x + 30

x = −2 ,

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7 x 2 + 3 x = 0

x = 0 , x = 3 7

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For the following exercises, solve the quadratic equation by using the square root property.

x 2 = 36

x = −6 , x = 6

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( x 1 ) 2 = 25

x = 6 , x = −4

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( 2 x + 1 ) 2 = 9

x = 1 , x = −2

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For the following exercises, solve the quadratic equation by completing the square. Show each step.

x 2 9 x 22 = 0

x = −2 , x = 11

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x 2 6 x = 13

x = 3 ± 22

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2 + z = 6 z 2

z = 2 3 , z = 1 2

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2 x 2 3 x 1 = 0

x = 3 ± 17 4

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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
Practice Key Terms 7

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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