verbal expression to quadratic expression
1.Find two consecutive odd numbers whose product is 23 more than their sum
2.The lenght of one leg of right triangle is 7 meters less than the lenght of the other leg. The lenght of hypotenuse is 13 meters. What are lenghts of the legs ? (Pythagorean Th.)
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Verified answer
1. Find two consecutive odd numbers whose product is 23 more than their sum.
Let the smaller odd number be x, and the next is x + 2.
x + (x + 2) + 23 = (x)(x + 2)
2x + 25 = x² + 2x
-x² + 25 = 0
x² - 25 = 0
(x + 5)(x - 5) = 0
x = -5, 5
The numbers are either 5 and 7 or -5 and -3.
2. The length of one leg of right triangle is 7 meters less than the length of the other leg. The length of hypotenuse is 13 meters. What are lengths of the legs?
a² + b² = c²
x² + (x - 7)² = (13)²
x² + x² - 14x + 49 = 169
2x² - 14x - 120 = 0
x² - 7x - 60 = 0
(x - 12)(x + 5) = 0
x = 12, -5
Since this is a measurement, use 12.
The legs are 12 and 7.