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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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