How to Make Quadratic Equations from Word Problems

Making quadratic equations from word problems can seem tricky, but there’s a simple way to do it! By following some easy steps, you can turn a real-life situation into a math equation. Here’s how to do it step by step:

Step 1: Read the Problem Carefully

• Understand the Problem: Start by reading the word problem. It’s important to grasp the whole situation.

• Identify the Topic: Figure out what the problem is about. Is it about something moving like a ball (projectile motion), measuring area, or figuring out profit/loss? Knowing this helps you see how the numbers relate.

Step 2: Define Your Variables

• Choose Your Variables: Pick letters to stand for the unknowns. You might use $h$ for height, $t$ for time, or $d$ for distance.

• Be Clear: Make sure each letter is explained in the problem's context. For example, if you’re looking at how high something goes over time, you’d say $h$ is height and $t$ is time.

Step 3: Identify Known Values and Relationships

• Gather Information: Look for numbers and relationships in the problem. This could mean starting points, coefficients, or values that relate to your variables.

• Spot Patterns: Quadratic equations often come up with certain topics, like areas (area = length × width) or physics (like how things move). Knowing these patterns helps when making your equation.

Step 4: Formulate the Equation

• Create the Quadratic Equation: Use the relationships and variables you identified to make your quadratic equation. Most quadratic equations look like this: $ax^2 + bx + c = 0$.

• Link Back to the Problem: Make sure your equation connects back to the situation described. For example, if it relates to area, it should show that clearly.

Step 5: Solve the Equation

• Pick How to Solve It: Decide how you’ll find the solutions based on your problem. You can use methods like factoring, completing the square, or the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Make Sense of the Results: After solving, think about what the solutions mean in relation to the original problem. If you end up with negative time, for instance, that doesn’t make sense in the real world.

Step 6: Validate the Solution

• Check Your Work: Make sure that your solution fits the original problem’s conditions. If there are two solutions, check that both make sense.

• Consider the Units: Look at whether the units (like meters or seconds) make sense and ensure the answers are relevant.

Examples of Making Quadratic Equations

Let’s look at two examples to see how to apply these steps.

Example 1: The Ball Thrown in the Air

Context: A ball is thrown from a 10-meter height. We can describe its height ($h$) after $t$ seconds using this equation:

$$h(t) = -5t^2 + 10t + 10$$

1. Understand it: The height will decrease because of gravity.

2. Define: Let $h$ = height and $t$ = time.

3. Identify Info: This shows the ball starts at 10 m high and how gravity affects it over time.

4. Create the Equation: We already have a quadratic equation here.

5. Solve: To find when the ball hits the ground, set $h(t) = 0$ and solve:

   $$-5t^2 + 10t + 10 = 0$$

   Use the quadratic formula or factoring here.

6. Check Your Work: Ensure the time you find is positive and makes sense.

Example 2: The Garden Area

Context: A rectangular garden must have an area of 120 square meters, and the length is 10 meters longer than the width.

1. Understand It: We know the area and a relationship between length and width.

2. Define Variables: Let width = $w$; therefore, length = $w + 10$.

3. Identify Values: The area is given as 120 m², an important fact.

4. Create the Equation: Set up this area equation:

   $$w(w + 10) = 120$$

   Expanding that gives:

   $$w^2 + 10w - 120 = 0$$

5. Solve: Use the quadratic formula to find $w$.

6. Check Your Work: Make sure both dimensions are positive and that the area is right.

Conclusion

By following these steps, you can easily turn different scenarios into quadratic equations. Understanding how to read the problem, define what you need, spot relationships, and set up the equations is vital for solving word problems with quadratics. The more you practice these steps, the better you’ll get at math! Always remember to check the results and ensure everything makes sense.