7.6 Quadratic equations (Page 3/9)

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Solve: $6 a^{2} + 9 a = 3 a$ .

$a = 0, a = −1$

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Solve: $45 b^{2} - 2 b = −17 b$ .

$b = 0, b = - \frac{1}{3}$

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Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?

Solve: $144 q^{2} = 25$ .

Solution

$\begin{matrix} \begin{matrix} Write the quadratic equation in standard form. \\ Factor. It is a difference of squares. \end{matrix} & \begin{matrix} 144 q^{2} & = & 25 \\ 144 q^{2} - 25 & = & 0 \\ (12 q - 5) (12 q + 5) & = & 0 \end{matrix} \\ \begin{matrix} Use the Zero Product Property to set each factor to 0 . \\ Solve each equation. \end{matrix} & \begin{matrix} 12 q - 5 & = & 0 & 12 q + 5 & = & 0 \\ 12 q & = & 5 & 12 q & = & −5 \\ q & = & \frac{5}{12} & q & = & - \frac{5}{12} \end{matrix} \\ Check your answers. \end{matrix}$

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Solve: $25 p^{2} = 49$ .

$p = \frac{7}{5}, p = - \frac{7}{5}$

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Solve: $36 x^{2} = 121$ .

$x = \frac{11}{6}, x = - \frac{11}{6}$

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The left side in the next example is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.

Solve: $(3 x - 8) (x - 1) = 3 x$ .

Solution

$\begin{matrix} \begin{matrix} Multiply the binomials. \\ Write the quadratic equation in standard form. \\ Factor the trinomial. \end{matrix} & \begin{matrix} (3 x - 8) (x - 1) & = & 3 x \\ 3 x^{2} - 11 x + 8 & = & 3 x \\ 3 x^{2} - 14 x + 8 & = & 0 \\ (3 x - 2) (x - 4) & = & 0 \end{matrix} \\ \begin{matrix} Use the Zero Product Property to set each factor to 0. \\ Solve each equation. \end{matrix} & \begin{matrix} 3 x - 2 & = & 0 & x - 4 & = & 0 \\ 3 x & = & 2 & x & = & 4 \\ x & = & \frac{2}{3} \end{matrix} \\ Check your answers. & The check is left to you! \end{matrix}$

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Solve: $(2 m + 1) (m + 3) = 12 m$ .

$m = 1, m = \frac{3}{2}$

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Solve: $(k + 1) (k - 1) = 8$ .

$k = 3, k = −3$

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The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree more than two by using the Zero Product Property, just like we solved quadratic equations.

Solve: $9 m^{3} + 100 m = 60 m^{2}$ .

Solution

$\begin{matrix} \begin{matrix} Bring all the terms to one side so that the other side is zero. \\ Factor the greatest common factor first. \\ Factor the trinomial. \end{matrix} & \begin{matrix} 9 m^{3} + 100 m & = & 60 m^{2} \\ 9 m^{3} - 60 m^{2} + 100 m & = & 0 \\ m (9 m^{2} - 60 m + 100) & = & 0 \\ m (3 m - 10) (3 m - 10) & = & 0 \end{matrix} \\ \begin{matrix} Use the Zero Product Property to set each factor to 0 . \\ Solve each equation. \end{matrix} & \begin{matrix} m & = & 0 & 3 m - 10 & = & 0 & 3 m - 10 & = & 0 \\ m & = & 0 & m & = & \frac{10}{3} & m & = & \frac{10}{3} \end{matrix} \\ Check your answers. & The check is left to you. \end{matrix}$

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Solve: $8 x^{3} = 24 x^{2} - 18 x$ .

$x = 0, x = \frac{3}{2}$

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Solve: $16 y^{2} = 32 y^{3} + 2 y$ .

$y = 0, y = \frac{1}{4}$

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When we factor the quadratic equation in the next example we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

Solve: $4 x^{2} = 16 x + 84$ .

Solution

$\begin{matrix} \begin{matrix} Write the quadratic equation in standard form. \\ Factor the greatest common factor first. \\ Factor the trinomial. \end{matrix} & \begin{matrix} 4 x^{2} & = & 16 x + 84 \\ 4 x^{2} - 16 x - 84 & = & 0 \\ 4 (x^{2} - 4 x - 21) & = & 0 \\ 4 (x - 7) (x + 3) & = & 0 \end{matrix} \\ \begin{matrix} Use the Zero Product Property to set each factor to 0. \\ Solve each equation. \end{matrix} & \begin{matrix} 4 & \neq & 0 & x - 7 & = & 0 & x + 3 & = & 0 \\ 4 & \neq & 0 & x & = & 7 & x & = & −3 \end{matrix} \\ Check your answers. & The check is left to you. \end{matrix}$

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Solve: $18 a^{2} - 30 = −33 a$ .

$a = - \frac{5}{2}, a = \frac{2}{3}$

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Solve: $123 b = −6 - 60 b^{2}$ .

$b = 2, b = \frac{1}{20}$

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Solve applications modeled by quadratic equations

The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.

Use a problem-solving strategy to solve word problems.

Read the problem. Make sure all the words and ideas are understood.
Identify what we are looking for.
Name what we are looking for. Choose a variable to represent that quantity.
Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

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

tijani

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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

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fine, how about you?

Mohammed

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

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