Quadratic Equation Word Problems Worksheet: Practice & Solutions
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Quadratic equations are an essential aspect of algebra that frequently appear in various real-life scenarios. From physics to engineering, understanding how to solve quadratic equations can help us analyze situations effectively. In this article, we'll delve into some quadratic equation word problems, and provide solutions to aid in your practice. Let's get started! 🚀
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation that can be expressed in the standard form:
[ ax^2 + bx + c = 0 ]
where:
- (a), (b), and (c) are constants,
- (x) is the variable,
- (a) cannot be equal to zero.
The solutions to a quadratic equation are known as the roots, and they can be found using various methods, including factoring, completing the square, or the quadratic formula:
[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} ]
Real-Life Applications of Quadratic Equations
Quadratic equations can be applied in numerous real-life situations:
- Projectile Motion: The height of a projectile can be modeled with a quadratic equation.
- Area Problems: Finding dimensions of rectangular areas when the area is given.
- Business: Maximizing profit or minimizing cost.
- Physics: Objects under the influence of gravity.
Word Problems Involving Quadratic Equations
Problem 1: Area of a Rectangle
A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 70 square meters, what are the dimensions of the garden?
Let:
- ( x ) = width of the garden (in meters)
- Length = ( x + 3 ) meters
Equation: [ x(x + 3) = 70 ] [ x^2 + 3x - 70 = 0 ]
Solution:
Using the quadratic formula, we can find the roots of the equation.
-
Identify ( a = 1 ), ( b = 3 ), and ( c = -70 ).
-
Calculate the discriminant: [ b^2 - 4ac = 3^2 - 4(1)(-70) = 9 + 280 = 289 ]
-
Apply the quadratic formula: [ x = \frac{{-3 \pm \sqrt{289}}}{2} ] [ x = \frac{{-3 \pm 17}}{2} ] This gives us two potential solutions:
- ( x = 7 ) (width)
- ( x = -10 ) (not valid)
Dimensions:
- Width = 7 meters
- Length = 10 meters (7 + 3)
Problem 2: Projectile Motion
A ball is thrown upward from a height of 1.5 meters with an initial velocity of 14 m/s. The height of the ball after ( t ) seconds is given by the equation:
[ h(t) = -4.9t^2 + 14t + 1.5 ]
Find out when the ball will hit the ground (i.e., when ( h(t) = 0 )).
Equation: [ -4.9t^2 + 14t + 1.5 = 0 ]
Solution:
-
Identify ( a = -4.9 ), ( b = 14 ), ( c = 1.5 ).
-
Calculate the discriminant: [ b^2 - 4ac = 14^2 - 4(-4.9)(1.5) = 196 + 29.4 = 225.4 ]
-
Apply the quadratic formula: [ t = \frac{{-14 \pm \sqrt{225.4}}}{2 \times -4.9} ] Calculating the roots gives us:
- Positive root: ( t \approx 3.1 ) seconds
- Negative root: Not applicable
Problem 3: Profit Maximization
A company determines that its profit ( P ) (in thousands of dollars) can be modeled by the equation:
[ P(x) = -2x^2 + 40x - 150 ]
where ( x ) is the number of units sold (in thousands). Find the number of units sold that maximizes profit.
Solution:
- Since the equation is in the standard form ( P(x) = ax^2 + bx + c ), we can find the vertex, which gives the maximum profit.
- The x-coordinate of the vertex is given by: [ x = \frac{{-b}}{{2a}} = \frac{{-40}}{{2(-2)}} = 10 ]
Maximum Profit: At 10,000 units sold, the maximum profit occurs.
Summary of Key Steps
| Problem | Equation Formulated | Dimensions/Result |
|---|---|---|
| Area of Rectangle | ( x^2 + 3x - 70 = 0 ) | Width = 7m, Length = 10m |
| Projectile Motion | ( -4.9t^2 + 14t + 1.5 = 0 ) | Hits the ground at t ≈ 3.1 s |
| Profit Maximization | ( P(x) = -2x^2 + 40x - 150 ) | Max profit at x = 10 |
Important Notes
"Quadratic equations not only provide theoretical understanding but are crucial in practical situations. Practice solving different types of word problems to strengthen your skills!"
Conclusion
Understanding and solving quadratic equations through word problems enhance your mathematical abilities and offer valuable insights into practical applications. By practicing with various types of problems, you’ll gain confidence in tackling quadratic equations, whether in academic settings or real-world applications. Remember, the key to mastering quadratic equations lies in consistent practice and familiarity with the various solving methods. Happy solving! 🎉