Problem 3 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

Solutions are \(t = 0\) and \(t = 2\).

Step 1: Rewrite the equation in standard form

Our task is to solve the quadratic equation \(t^2 = 2t\). First, rewrite this equation in standard form, which is \(ax^2 + bx + c = 0\). We do this by subtracting \(2t\) from both sides to get \(t^2 - 2t = 0\).

Step 2: Factor the quadratic expression

Now, factor the quadratic expression \(t^2 - 2t\). The common factor in both terms is \(t\), so factor \(t\) out: \(t(t - 2) = 0\).

Step 3: Apply the Zero Product Property

The Zero Product Property states that if the product of two terms is zero, at least one of the terms must be zero. Therefore, set each factor equal to zero: \(t = 0\) or \(t - 2 = 0\).

Step 4: Solve each equation from the factors

Solve each equation separately:1. For \(t = 0\), the solution is plainly \(t = 0\).2. For \(t - 2 = 0\), add 2 to both sides to get \(t = 2\). Thus, the solutions are \(t = 0\) and \(t = 2\).

Step 5: Check the solutions

Substitute each solution back into the original equation \(t^2 = 2t\) to verify correctness:1. For \(t = 0\), \(0^2 = 0\) and \(2(0) = 0\), so both sides are equal.2. For \(t = 2\), \(2^2 = 4\) and \(2(2) = 4\), so both sides are equal. Both solutions check out and satisfy the original equation.

Key Concepts

Factoring Quadratics

Factoring quadratics is a key method for solving quadratic equations, where the equation is expressed in the form \(ax^2 + bx + c = 0\). It involves rewriting the equation so that it is easier to solve. The first step is to ensure the equation is in its standard form. This can involve rearranging terms or performing algebraic operations to isolate all terms on one side of the equation.
For example, consider the quadratic equation \(t^2 = 2t\). To factor this equation, we first rewrite it as \(t^2 - 2t = 0\). This standard form makes it clear what factors to look for. Once in standard form, notice if there are common factors for all the terms.
A common factor is a number or variable that divides evenly into each term. In this case, both \(t^2\) and \(-2t\) share the factor \(t\). By factoring \(t\) out, we get \(t(t - 2) = 0\). This expression is easier to work with because it's been broken into the product of two simpler terms.

Zero Product Property

The Zero Product Property is a fundamental concept used once a quadratic has been factored into simpler terms. It states that if a product of two factors equals zero, then at least one of the factors must be zero. This property makes it straightforward to find the solutions to a quadratic equation.
After factoring \(t^2 - 2t\) into \(t(t - 2) = 0\), apply this property. Set each factor equal to zero:

• First, \(t = 0\)
• Second, \(t - 2 = 0\)

This approach reduces the problem to solving simple linear equations, each set equal to zero. The Zero Product Property effectively splits the problem into parts that are easier to handle, leading directly to the solutions of the original quadratic equation.

Solving Algebraic Equations

Solving algebraic equations involves finding the values of the unknowns that satisfy the equation. When solving the factored form of a quadratic equation, the tasks become easier and more methodical. Each factor, set to zero, gives us a simple linear equation.
These linear equations stem from the application of the Zero Product Property. For example, from \(t(t - 2) = 0\) where \(t = 0\) and \(t - 2 = 0\), solve both:

• From \(t = 0\), the solution is simply \(t = 0\).
• From \(t - 2 = 0\), add 2 to both sides to solve for \(t\), which yields \(t = 2\).

Verifying the solutions by substitution back into the original equation ensures accuracy. Substituting both solutions back into \(t^2 = 2t\), you confirm that both \(t = 0\) and \(t = 2\) satisfy the equation, completing the solution process. This verification strengthens your understanding and mastery of solving quadratic equations through factoring and the zero product property.