Problem 2 Applying the Quadratic Formula

GRIDDED RESPONSE

Fundraising Your school's jazz band is selling CDs as a fundraiser. The total profit p depends on the amount x that your band charges for each CD. The equation p = − x 2 + 48 x − 300 bold italic p equals negative , bold italic x squared , plus 48 bold italic x minus 300  models the profit of the fundraiser. What is the least amount, in dollars, you can charge for a CD to make a profit of $200?

p = − x 2 + 48 x − 300   200 = − x 2 + 48 x − 300 Substitute  200  for  p . 0 = − x 2 + 48 x − 500 Write the equation in standard form . a = − 1 , b = 48 , c = − 500 Find the values of  a , b , and  c . x = − 48 ± 48 2 − 4 ( − 1 ) ( − 500 ) 2 ( − 1 ) Substitute into − b ± b 2 − 4 a c 2 a . x = − 48 ± 304 − 2 Simplify . x ≈ 15.282  or  x ≈ 32.717 Use a calculator . table with 7 rows and 3 columns , row1 column 1 , p , column 2 equals negative , x squared , plus 48 x minus 300 , column 3 , row2 column 1 , 200 , column 2 equals negative , x squared , plus 48 x minus 300 , column 3 cap substitute . 200 , for , p . , row3 column 1 , 0 , column 2 equals negative , x squared , plus 48 x minus 500 , column 3 cap writetheequationinstandardform . . , row4 column 1 , eh , column 2 equals negative 1 comma b equals 48 comma c equals negative 500 , column 3 cap findthevaluesof . eh comma b comma , and , c . , row5 column 1 , x , column 2 equals . fraction negative 48 plus minus . square root of 48 squared , minus 4 . open , negative 1 , close . open , negative 500 , close end root , over 2 . open , negative 1 , close end fraction , column 3 cap substituteinto . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction . . , row6 column 1 , x , column 2 equals . fraction negative 48 plus minus , square root of 304 , over negative 2 end fraction , column 3 cap simplify , . , row7 column 1 , x , column 2 almost equal to , 15.282 . or , x almost equal to , 32.717 , column 3 cap useacalculator . . , end table

The least amount you can charge is $15.29 for each CD to make a profit of $200.

The answer is 15.29.

✓ Got It?

2. 1. In Problem 2, what is the least amount you can charge for each CD to make a $100 profit?
   2. Reasoning Would a negative profit make sense in this problem? Explain.

Take Note Key Concept: Discriminant

The discriminant of a quadratic equation in the form a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0  is the value of the expression b 2 − 4 a c . b squared , minus 4 eh c .

x = − b ± b 2 − 4 a x 2 a ← discriminant x equals . fraction negative b plus minus . square root of b squared , minus 4 eh x end root , over 2 eh end fraction . leftwards arrow . discriminant