Problem 139 - Elementary Algebra

Question

A flare is fired straight up from a ship at sea. Solve the equation \(16\left(t^{2}-13 t+40\right)=0\) for \(t,\) the number of seconds it will take for the flare to be at an altitude of 640 feet.

Step-by-Step Solution

Answer

The solutions for t are 5 seconds and 8 seconds.

Step 1: Simplify the given equation

Start by simplifying the given equation. You have the equation:

Step 2: Divide both sides by 16

To simplify, divide both sides of the equation by 16:

Step 3: Solve the simplified quadratic equation

Now, solve the quadratic equation:

Step 4: Factorize the quadratic equation

Factorize the quadratic equation:

Step 5: Find the values of t

Set each factor equal to zero and solve for t:

Step 6: Verify the solutions

Verify the solutions by substituting the values back into the original equation (optional):

Key Concepts

factoring

Factoring turns a quadratic equation into a simple product of expressions. In the example, we have to factorize the quadratic expression from the simplified equation. To factor, find two numbers that multiply to the constant term (in this case, 40) and add up to the middle term coefficient (13). Using this method helps to easily break down and solve quadratic equations. Always check different factor combinations until you find the one that fits.

solving quadratic equations

Solving quadratic equations involves finding the values of the variable that satisfy the equation. We follow these steps:

• Start by simplifying the equation if needed.
• Next, divide both sides by a common factor to make the equation easier to work with.
• Then factorize the quadratic expression.
• Set each factor equal to zero.
• Solve for the variable.

Using these steps ensures we systematically find all possible solutions for the quadratic equation.

verifying solutions

Verifying solutions ensures that the proposed solutions satisfy the original equation. Substitute each found value of the variable back into the original quadratic equation. In our example, check if substituting the values of t simplifies the equation to zero. If it does, the solutions are correct. This step helps to confirm our work and ensures we haven't missed any mistakes.

mathematical problem-solving

Mathematical problem-solving involves understanding the problem, devising a plan, carrying out the plan, and reviewing the results. With quadratic equations:

• Start by understanding the structure of the given equation.
• Simplify and reduce the problem step by step to make it manageable.
• Solve for the variable and verify your solutions to ensure accuracy.
• Reflect on the process to understand what worked and what didn’t for future improvements.

Practicing these steps builds confidence and aids in tackling more complex problems.