Problem 48 - Algebra for College Students

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: Rewrite the Equation

Start by rewriting the given equation in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). The given equation is \(x^2 = 16x\). Move all terms to one side of the equation to get \(x^2 - 16x = 0\).

Step 2: Factor the Expression

The equation \(x^2 - 16x = 0\) can be factored by taking out the common factor \(x\). So, the equation becomes \(x(x - 16) = 0\).

Step 3: Apply the Zero Product Property

Use the zero product property, which states that if \(ab = 0\), then \(a = 0\) or \(b = 0\). This gives us two equations: \(x = 0\) and \(x - 16 = 0\). Solve each equation separately.

Step 4: Solve Each Equation

Solve the first equation: \(x = 0\).Solve the second equation: \(x - 16 = 0\) which gives \(x = 16\).

Step 5: Write the Solution

The solutions to the quadratic equation \(x^2 = 16x\) are \(x = 0\) and \(x = 16\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential algebraic technique used to simplify and solve quadratic equations. A quadratic equation typically takes the standard form of \(ax^2 + bx + c = 0\). In the exercise, converting the given equation \(x^2 = 16x\) to a standard form yields \(x^2 - 16x = 0\).

Factoring involves finding two terms that multiply together to give the original quadratic equation. For \(x^2 - 16x = 0\), we notice that both terms share a common factor: \(x\). By pulling out this common factor, the equation becomes \(x(x - 16) = 0\).

Factoring not only simplifies the problem but also sets the stage for the next step in solving the equation by allowing us to use the Zero Product Property.

Zero Product Property

The Zero Product Property is a crucial principle in algebra that helps solve equations after factoring. It states that if a product of two numbers is zero, then at least one of the numbers must be zero. Using the property is straightforward: given \(ab = 0\), either \(a = 0\) or \(b = 0\).

In the equation \(x(x - 16) = 0\), you apply this property directly. This means that either \(x = 0\) or \(x - 16 = 0\).

• For \(x = 0\), the solution is immediate.
• For \(x - 16 = 0\), rearrange to find \(x = 16\).

Applying the Zero Product Property is an efficient way to find solutions in quadratic equations once they have been properly factored.

Solving Quadratic Equations

Solving quadratic equations involves finding values of \(x\) that make the equation true. The manual steps include rewriting the equation in standard form, factoring, applying the Zero Product Property, and finally solving the resulting simple equations.

The conversion of the initial quadratic equation \(x^2 = 16x\) to \(x^2 - 16x = 0\) set the stage for solving. Once the quadratic was factored to \(x(x - 16) = 0\), solving became straightforward by using the Zero Product Property.

• Confirming that \(x = 0\) is a solution is as simple as substituting back into the original equation to check that it holds true.
• Similarly, for \(x = 16\), substitution verifies its correctness.

These solutions, \(x = 0\) and \(x = 16\), are exactly how quadratic equations can often result in two answers, capturing the essence of solving quadratics thoroughly.