# 2.5 Quadratic equations  (Page 7/14)

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## Algebraic

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

${x}^{2}+4x-21=0$

${x}^{2}-9x+18=0$

$x=6,$ $x=3$

$2{x}^{2}+9x-5=0$

$6{x}^{2}+17x+5=0$

$x=\frac{-5}{2},$ $x=\frac{-1}{3}$

$4{x}^{2}-12x+8=0$

$3{x}^{2}-75=0$

$x=5,$ $x=-5$

$8{x}^{2}+6x-9=0$

$4{x}^{2}=9$

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

$2{x}^{2}+14x=36$

$5{x}^{2}=5x+30$

$x=-2,$

$4{x}^{2}=5x$

$7{x}^{2}+3x=0$

$x=0,$ $x=\frac{-3}{7}$

$\frac{x}{3}-\frac{9}{x}=2$

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

${x}^{2}=36$

$x=-6,$ $x=6$

${x}^{2}=49$

${\left(x-1\right)}^{2}=25$

$x=6,$ $x=-4$

${\left(x-3\right)}^{2}=7$

${\left(2x+1\right)}^{2}=9$

$x=1,$ $x=-2$

${\left(x-5\right)}^{2}=4$

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

${x}^{2}-9x-22=0$

$x=-2,$ $x=11$

$2{x}^{2}-8x-5=0$

${x}^{2}-6x=13$

$x=3±\sqrt{22}$

${x}^{2}+\frac{2}{3}x-\frac{1}{3}=0$

$2+z=6{z}^{2}$

$z=\frac{2}{3},$ $z=-\frac{1}{2}$

$6{p}^{2}+7p-20=0$

$2{x}^{2}-3x-1=0$

$x=\frac{3±\sqrt{17}}{4}$

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$2{x}^{2}-6x+7=0$

${x}^{2}+4x+7=0$

Not real

$3{x}^{2}+5x-8=0$

$9{x}^{2}-30x+25=0$

One rational

$2{x}^{2}-3x-7=0$

$6{x}^{2}-x-2=0$

Two real; rational

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution .

$2{x}^{2}+5x+3=0$

${x}^{2}+x=4$

$x=\frac{-1±\sqrt{17}}{2}$

$2{x}^{2}-8x-5=0$

$3{x}^{2}-5x+1=0$

$x=\frac{5±\sqrt{13}}{6}$

${x}^{2}+4x+2=0$

$4+\frac{1}{x}-\frac{1}{{x}^{2}}=0$

$x=\frac{-1±\sqrt{17}}{8}$

## Technology

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x -intercepts) by using 2 nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

${\text{Y}}_{1}=4{x}^{2}+3x-2$

${\text{Y}}_{1}=-3{x}^{2}+8x-1$

$x\approx 0.131\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\approx 2.535$

${\text{Y}}_{1}=0.5{x}^{2}+x-7$

To solve the quadratic equation $\text{\hspace{0.17em}}{x}^{2}+5x-7=4,$ we can graph these two equations

$\begin{array}{l}\hfill \\ \begin{array}{l}{\text{Y}}_{1}={x}^{2}+5x-7\hfill \\ {\text{Y}}_{2}=4\hfill \end{array}\hfill \end{array}$

and find the points of intersection. Recall 2 nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

$x\approx -6.7\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\approx 1.7$

To solve the quadratic equation $\text{\hspace{0.17em}}0.3{x}^{2}+2x-4=2,$ we can graph these two equations

$\begin{array}{l}\hfill \\ \begin{array}{l}{\text{Y}}_{1}=0.3{x}^{2}+2x-4\hfill \\ {\text{Y}}_{2}=2\hfill \end{array}\hfill \end{array}$

and find the points of intersection. Recall 2 nd CALC 5:intersection. Do this and find the solutions to the nearest tenth.

## Extensions

Beginning with the general form of a quadratic equation, $\text{\hspace{0.17em}}a{x}^{2}+bx+c=0,$ solve for x by using the completing the square method, thus deriving the quadratic formula.

$\begin{array}{ccc}\hfill a{x}^{2}+bx+c& =& 0\hfill \\ \hfill {x}^{2}+\frac{b}{a}x& =& \frac{-c}{a}\hfill \\ \hfill {x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}& =& \frac{-c}{a}+\frac{b}{4{a}^{2}}\hfill \\ \hfill {\left(x+\frac{b}{2a}\right)}^{2}& =& \frac{{b}^{2}-4ac}{4{a}^{2}}\hfill \\ \hfill x+\frac{b}{2a}& =& ±\sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}\hfill \\ \hfill x& =& \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}$

Show that the sum of the two solutions to the quadratic equation is $\text{\hspace{0.17em}}\frac{-b}{a}.$

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft. 2 . Solve the quadratic equation to find the length and width.

$x\left(x+10\right)=119;$ 7 ft. and 17 ft.

Abercrombie and Fitch stock had a price given as $\text{\hspace{0.17em}}P=0.2{t}^{2}-5.6t+50.2,$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the time in months from 1999 to 2001. ( $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ is January 1999). Find the two months in which the price of the stock was $30. Suppose that an equation is given $\text{\hspace{0.17em}}p=-2{x}^{2}+280x-1000,$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the number of items sold at an auction and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2 nd CALC maximum. To obtain a good window for the curve, set $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ [0,200] and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ [0,10000]. maximum at $\text{\hspace{0.17em}}x=70$ ## Real-world applications A formula for the normal systolic blood pressure for a man age $\text{\hspace{0.17em}}A,$ measured in mmHg, is given as $\text{\hspace{0.17em}}P=0.006{A}^{2}-0.02A+120.\text{\hspace{0.17em}}$ Find the age to the nearest year of a man whose normal blood pressure measures 125 mmHg. The cost function for a certain company is $\text{\hspace{0.17em}}C=60x+300\text{\hspace{0.17em}}$ and the revenue is given by $\text{\hspace{0.17em}}R=100x-0.5{x}^{2}.\text{\hspace{0.17em}}$ Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of$300.

The quadratic equation would be $\text{\hspace{0.17em}}\left(100x-0.5{x}^{2}\right)-\left(60x+300\right)=300.\text{\hspace{0.17em}}$ The two values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ are 20 and 60.

A falling object travels a distance given by the formula $\text{\hspace{0.17em}}d=5t+16{t}^{2}\text{\hspace{0.17em}}$ ft, where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. How long will it take for the object to traveled 74 ft?

A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft 2 . Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.

3 feet

An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, $\text{\hspace{0.17em}}P,$ who contracted the flu $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ days after it broke out is given by the model $\text{\hspace{0.17em}}P=-{t}^{2}+13t+130,$ where $\text{\hspace{0.17em}}1\le t\le 6.\text{\hspace{0.17em}}$ Find the day that 160 students had the flu. Recall that the restriction on $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is at most 6.

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(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
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how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
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given that x= 3/5 find sin 3x
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remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
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(x2-2x+8)-4(x2-3x+5)
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Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
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x²-2x+9-4x²+12x-20 -3x²+10x+11
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(X2-2X+8)-4(X2-3X+5)=0 ?
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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
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Soo sorry (5±Root11* i)/3
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Mukhtar
2x²-6x+1=0
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explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
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Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
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A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
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What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
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a function
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a function
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given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
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Miranda
I am living in india
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what is the formula for calculating algebraic
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
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