Problem 26 - Elementary Algebra

Question

Solve. $$ k(k+10)(k+10)=0 $$

Step-by-Step Solution

Answer

The solutions are \(k = 0\) and \(k = -10\).

Step 1: Identify the equation

The equation given is: \[k(k+10)(k+10) = 0\]

Step 2: Simplify the equation

Recognize that \[(k+10)(k+10)\] can be simplified to \[(k+10)^2\]. Hence, the equation becomes: \[k(k+10)^2 = 0\]

Step 3: Apply the zero-product property

According to the zero-product property, if the product of factors is zero, at least one of the factors must be zero. Therefore, set each factor equal to zero: \[k = 0\] or \[(k + 10)^2 = 0\]

Step 4: Solve each factor

First, solve for \[k = 0\] This gives one solution: \(k = 0\). Next, solve the equation \[(k + 10)^2 = 0\]. Taking the square root of both sides gives: \[k + 10 = 0\]. Then, subtract 10 from both sides to find: \(k = -10\).

Step 5: Write the final solution

The solutions to the equation are \(k = 0\) and \(k = -10\).

Key Concepts

Zero-Product Property

The zero-product property is a fundamental concept in algebra that states if the product of two or more factors is zero, at least one of the factors must be zero. This principle is crucial for solving polynomial equations. For example, in the equation \[ k(k+10)(k+10) = 0 \], we can use the zero-product property to solve for \[ k \]. Once we simplify the equation to \[ k(k+10)^2 = 0 \], we recognize that the equation will hold true if either \[ k = 0 \] or the factor \[ (k+10)^2 = 0 \]. This allows us to set up separate equations and find all possible roots.

Factoring

Factoring is the process of breaking down an expression into simpler terms (or factors) that, when multiplied together, give the original expression. In our example, \[ k(k+10)(k+10) = 0 \], the expression is already partially factored. Recognizing that \[ (k+10)(k+10) \] can be rewritten as \[ (k+10)^2 \], we further simplify the equation to \[ k(k+10)^2 = 0 \]. Factoring is a useful skill because it allows us to apply the zero-product property and solve for the variable more easily.

Simplifying Equations

Simplifying equations makes them easier to solve. It often involves combining like terms, reducing fractions, and identifying patterns. In the given problem, simplifying \[ (k+10)(k+10) \] to \[ (k+10)^2 \] is a key step. This simplification reduces the complexity of the equation, making it straightforward to apply the zero-product property. Simplifying equations not only aids in solving them but also helps in understanding the relationship between the terms.

Solutions to Polynomial Equations

Polynomial equations can have multiple solutions based on the degree of the polynomial. In our example, the simplified equation \[ k(k+10)^2 = 0 \] is a polynomial equation of degree 3. Using the zero-product property, we set each factor equal to zero: \[ k = 0 \] and \[ (k + 10)^2 = 0 \]. Solving these, we get \[ k = 0 \] and \[ k = -10 \]. There are two distinct solutions: \[ k = 0 \] and \[ k = -10 \]. Understanding how to solve polynomial equations is key to finding all possible values of the variable that satisfy the equation.