Problem 4 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ t^{2}-7 t=0 $$

Step-by-Step Solution

Answer

The solutions are \( t = 0 \) and \( t = 7 \).

Step 1: Set the Equation to Zero

We start with the quadratic equation: \( t^2 - 7t = 0 \). This is already equal to zero, so we can move on to the next step.

Step 2: Factor the Equation

Factor the left side of the equation. \( t^2 - 7t \) can be factored by taking \( t \) as a common factor: \( t(t - 7) = 0 \).

Step 3: Set Each Factor to Zero

To find the solutions, set each factor equal to zero: \( t = 0 \) and \( t - 7 = 0 \).

Step 4: Solve Each Equation for t

Solve each equation to find the possible values for \( t \). For \( t = 0 \), the solution is \( t = 0 \). For \( t - 7 = 0 \), add 7 to both sides to get \( t = 7 \).

Step 5: Check Your Answers

Plug each solution back into the original equation to verify correctness. For \( t = 0 \), substitute to get \( 0^2 - 7(0) = 0 \), which is true. For \( t = 7 \), substitute to get \( 7^2 - 7(7) = 0 \), which is also true.

Key Concepts

Factoring Quadratic Equations

Factoring is a key step in solving quadratic equations. It involves finding two expressions that multiply together to get the original equation. In our example, the equation is \( t^2 - 7t = 0 \). To factor it, we look for common factors.
Here, \( t \) is a common factor on the left side, which gives us \( t(t - 7) = 0 \).
When factoring a quadratic, always try to take out the greatest common factor first. After removing shared terms, you simplify the expression into two separate factors. If you multiply these factors, you will get back the original equation.
Factoring is essential because it simplifies the equation, making it easier to find solutions. It breaks a problem into smaller, manageable parts, simplifying further steps.

Solutions of Quadratic Equations

Finding solutions to quadratic equations involves setting each factor equal to zero and solving them separately. In our exercise, the factored form is \( t(t - 7) = 0 \).
Each factor can potentially be a solution to the equation.

• Set \( t = 0 \), which directly provides one solution: \( t = 0 \).
• Set \( t - 7 = 0 \). Solve it by adding 7 to both sides, resulting in another solution: \( t = 7 \).

These solutions, \( t = 0 \) and \( t = 7 \), are the roots of the quadratic equation. They represent the points where the parabola described by the quadratic equation crosses the horizontal axis in a graph.
It's important to note that solutions are found by solving each factor separately and ensuring all possibilities in the factored expression are explored.

Checking Solutions

Once you've found solutions to a quadratic equation, always verify their correctness by substituting them back into the original equation. This is crucial for confirming that your solutions are true. Let's see how this is done.
For the solution \( t = 0 \), substitute it back into the original equation: \( 0^2 - 7(0) = 0 \). This equation holds true, meaning \( t = 0 \) is a valid solution.
Now, for \( t = 7 \), substitute back similarly: \( 7^2 - 7(7) = 0 \). Simplifying the left side, we see it also yields zero, confirming that \( t = 7 \) is correct.
Checking solutions is a final step that ensures no mistakes were made in earlier steps. It reassures you that the solutions make the equation true when plugged back into it. This step of verification builds confidence and ensures accuracy in solving other mathematical problems as well.