2.5 Quadratic equations  (Page 7/14)

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

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

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

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

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

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

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

$x^2+x=4$

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

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

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

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

${\text{Y}}_1=4x^2+3x-2$

${\text{Y}}_1=\mathrm{-3}{x}^2+8x-1$

${\text{Y}}_1=0.5x^2+x-7$

To solve the quadratic equation $x^2+5x-7=4,$ 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.

To solve the quadratic equation $0.3x^2+2x-4=2,$ we can graph these two equations

$\begin{array}{l}\hfill \\ \begin{array}{l}{\text{Y}}_1=0.3x^2+2x-4\hfill \\ {\text{Y}}_2=2\hfill \end{array}\hfill \end{array}$

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

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

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

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.

Abercrombie and Fitch stock had a price given as $P=0.2t^2-5.6t+50.2,$ where $t$ is the time in months from 1999 to 2001. ( $t=1$ is January 1999). Find the two months in which the price of the stock was $30.

Suppose that an equation is given $p=\mathrm{-2}{x}^2+280x-1000,$ where $x$ represents the number of items sold at an auction and $p$ 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 $x$ [0,200] and $y$ [0,10000].

A formula for the normal systolic blood pressure for a man age $A,$ measured in mmHg, is given as $P=0.006A^2-0.02A+120.$ 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 $C=60x+300$ and the revenue is given by $R=100x-0.5x^2.$ 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.

A falling object travels a distance given by the formula $d=5t+16t^2$ ft, where $t$ 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.

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