# 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.

what is the period of cos?
Patrick
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what is the formula used for this question? "Jamal wants to save \$54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
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log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
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1×10^0
Akugry
Evalute exponential functions
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom