Translate real-world word problems into quadratic equations and solve them, covering ages, numbers, speed, and geometric contexts.
Word problems about numbers or ages often translate directly into quadratic equations once relationships are expressed algebraically.
The product of two consecutive positive integers is 90. Find the integers.
x(x+1)=90; x²+x−90=0; (x+10)(x−9)=0; x=9 (positive); integers are 9,10
A number is 5 more than another, and their product is 84. Find the numbers.
x(x+5)=84; x²+5x−84=0; (x+12)(x−7)=0; x=7; numbers are 7,12
Problems involving speed, time, and distance often lead to quadratic equations when time is affected by a change in speed.
A train travels 300 km at a certain speed. If the speed had been 5 km/h more, it would take 2 hours less. Find the original speed.
300/x − 300/(x+5)=2; solving gives x=25 km/h
Why do speed problems often produce quadratic equations?
Because time=distance/speed, and when speed changes, cross-multiplying the time difference produces a squared term
1. The product of two consecutive positive integers is 132. Find the integers.
2. A number is 3 more than another, and their product is 40. Find the numbers.
3. The sum of the squares of two consecutive positive integers is 61. Find the integers.
4. A rectangular field has area 96 m² and length 4m more than its breadth. Find the dimensions.
5. A train covers 180 km at a certain speed. If speed were 6 km/h more, time taken would be 1 hour less. Find the speed.