Problem 37 - Beginning and Intermediate Algebra

Question

Solve each equation. $$p^{2}=11 p$$

Step-by-Step Solution

Answer

The short answer to the given equation \(p^2 = 11p\) is: \(p = 0\) and \(p = 11\).

Step 1: Rearrange the equation to set it equal to 0

Subtract 11p from both sides of the equation to set it equal to 0: $$p^2 - 11p = 0$$

Step 2: Factor the equation

If possible, factor out any common factors to simplify the equation. In this case, we can factor out a p: $$p(p - 11) = 0$$

Step 3: Solve the factored equation for p

Now that the equation is factored, we can set each factor equal to 0 and solve for p. This will give us the solutions to the original equation. $$p = 0$$ and $$p - 11 = 0 \Rightarrow p = 11$$

Step 4: Check the solutions

Check the solutions by plugging them back into the original equation to ensure that they are correct: $$p^2 = 11p$$ For p=0: $$(0)^2 = 11(0)$$ $$0 = 0$$ For p=11: $$(11)^2 = 11(11)$$ $$121 = 121$$ Both solutions are correct. So, the solutions to the equation are: $$p = 0 \ \text{and} \ p = 11$$

Key Concepts

factoring quadratic equations

Factoring quadratic equations is an essential skill in algebra that helps simplify and solve quadratics. A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \). To factor means to write the equation as a product of its factors, essentially reversing the process of multiplication.
For example, consider the equation \( p^2 - 11p = 0 \). The first step in factoring is to look for common factors in the terms of the equation. Here, both terms contain a \( p \), so we can factor it out:

• \( p(p - 11) = 0 \)

Factoring breaks the quadratic into linear components, making it easier to solve. It's critical to ensure that the equation is simplified as much as possible by factoring out common terms.

solving quadratic equations

Once a quadratic is factored, solving it becomes more straightforward. The Zero Product Property states that if a product of two numbers is zero, then at least one of the numbers must be zero. Hence, after factoring \( p(p - 11) = 0 \), we set each factor equal to zero:

• \( p = 0 \)
• \( p - 11 = 0 \)

Solve each equation separately to find the values of \( p \). This property turns the process of solving a quadratic equation into solving simple linear equations. It reduces complexity and provides a clear pathway to finding the solutions.

algebraic solutions

Finding algebraic solutions involves manipulating the equation to isolate the variable. Solving true algebraic solutions means finding all possible values that satisfy the original equation. For \( p^2 - 11p = 0 \), we obtained the solutions by factoring and solving

• \( p = 0 \)
• \( p = 11 \)

Each solution corresponds to a point where the original quadratic function crosses the x-axis, mathematically described as the roots of the equation.

check solutions in algebra

Checking solutions is a crucial step in solving equations, as it confirms their validity. After solving \( p = 0 \) and \( p = 11 \), we substitute each back into the original equation \( p^2 = 11p \) to verify correctness.
For \( p = 0 \):

• \( 0^2 = 11 imes 0 \)
• The equation holds as \( 0 = 0 \).

For \( p = 11 \):

• \( 11^2 = 11 imes 11 \)
• The equation holds as \( 121 = 121 \).

Checking ensures that no computational errors influenced the final result and confirms that both solutions satisfy the original equation.