이차방정식의 판별식과 근과 계수의 관계

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 $P=0.2{t}^{2}-5.6t+50.2,$ 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 $p=-2{x}^{2}+280x-1000,$ 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 $P=0.006{A}^{2}-0.02A+120.$ 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 $R=100x-0.5{x}^{2}.$ 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 $d=5t+16{t}^{2}$ ft, where $t$ is measured in seconds. How long will it take for the object to travel 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 ft². 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.

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

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

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

$3{x}^{2}+18x+15=0$

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

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

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

$7{x}^{2}-9x=0$

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

${x}^{2}=49$

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

${(x-4)}^{2}=36$

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

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

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

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

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

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

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

$15{x}^{2}-x-2=0$