Problem 48 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate. $$ x^{2}=16 x $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = 16 \).

Step 1: Recognize the Form of the Equation

The given equation is \( x^2 = 16x \). The equation is in a standard quadratic form if rewritten as \( x^2 - 16x = 0 \). This equation qualifies for solving by factoring as it is set to zero.

Step 2: Factor the Equation

To factor the equation \( x^2 - 16x = 0 \), first look for a common factor in both terms. Both terms have \( x \) as a common factor, so factor out \( x \) to get \( x(x - 16) = 0 \).

Step 3: Apply the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Apply this property to \( x(x - 16) = 0 \) which gives us two equations: \( x = 0 \) and \( x - 16 = 0 \).

Step 4: Solve Each Equation

Solve the equations obtained from the Zero Product Property:- For \( x = 0 \), the solution is \( x = 0 \).- For \( x - 16 = 0 \), add 16 to both sides to find \( x = 16 \).

Step 5: Verify the Solutions

Lastly, verify the solutions by substituting them back into the original equation \( x^2 = 16x \):- For \( x = 0 \), \( 0^2 = 0 \cdot 16 \), which satisfies the equation.- For \( x = 16 \), \( 16^2 = 16 \cdot 16 \), which satisfies the equation.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a powerful technique in algebra. It allows us to break down quadratic expressions into simpler factors that, when multiplied together, give the original equation. This method is often used when the quadratic equation is set to zero, making it easier to solve. For instance, in the equation \( x^2 - 16x = 0 \), we look for terms that are common across both parts of the expression. In this case, the variable \( x \) is present in both terms. By factoring out \( x \), the equation simplifies to \( x(x - 16) = 0 \).

• Identify common factors shared by both terms.
• Factor out these shared terms to simplify the equation.

This makes the equation ready for the next step – applying the zero product property.

Zero Product Property

The zero product property is a fundamental concept that states if the product of two numbers is zero, then at least one of those numbers must be zero. This property is extremely useful when solving factored quadratic equations. Once you have factored an equation, like \( x(x - 16) = 0 \), you can divide it into two separate equations:

• \( x = 0 \)
• \( x - 16 = 0 \)

The reasoning here is straightforward—since their product is zero, one or both of them must equal zero to satisfy the equation. This step is crucial in narrowing down the potential solutions for \( x \), as it effectively breaks the problem into smaller, simpler equations that are easily solved.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. After applying factoring and the zero product property, you will have simple linear equations to solve.

• For \( x = 0 \), no further computation is needed. This is already a solution.
• For \( x - 16 = 0 \), add 16 to both sides to solve for \( x \). This gives \( x = 16 \).

It's important to verify each solution by plugging them back into the original equation \( x^2 = 16x \). This check confirms whether the solutions satisfy the initial equation:

• Substituting \( x = 0 \), you get \( 0^2 = 0 \cdot 16 \), which holds true.
• Substituting \( x = 16 \), you get \( 16^2 = 16 \cdot 16 \), which is also correct.

This entire process ensures that the solutions are accurate and fulfill the conditions of the original equation.