# 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.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin