Problem 13 - Algebra 2

Question

Solve each equation by factoring or by taking square roots. $$ x^{2}-4 x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = 4\).

Step 1: Factor the quadratic equation

First, notice that both terms on the left-hand side of the equation have a common factor of 'x'. Factor out the 'x' from both terms: $$ x(x - 4) = 0 $$

Step 2: Apply the Zero Product Property

Since the product of two factors equals zero, at least one of the factors must be zero. Set each factor equal to zero and solve for 'x': $$ x = 0 $$$$ x - 4 = 0 $$

Step 3: Solve for x

Solve the individual equations from the previous step: For the first equation, nothing more is needed since 'x' is already isolated: $$ x = 0 $$ For the second equation, add 4 to both sides of the equation to isolate 'x': $$ x - 4 + 4 = 0 + 4 $$$$ x = 4 $$

Key Concepts

Factoring Quadratics

The process of factoring quadratics involves expressing a quadratic equation in the form of \( ax^2 + bx + c = 0 \) as a product of its linear factors. This can make solving for the variable much easier. First, identify any common factors among the terms and factor them out. Then, look for two numbers that multiply to give the constant term (\( c \) in the standard form) and add up to the coefficient of the middle term (\( b \) in the standard form). For the equation \( x^2 - 4x = 0 \) from our exercise, the common factor is 'x'. Once factored, we get \( x(x - 4) = 0 \).

Understanding this concept is crucial as it simplifies the problem into more manageable chunks. Factoring is often the first step we take towards finding the solutions of a quadratic equation.

Zero Product Property

The Zero Product Property states that if the product of two expressions is zero, then at least one of the expressions must also be zero. This is a fundamental principle in algebra and can be applied to solve equations where the expression is factored into multiple terms. For instance, after factoring our quadratic equation, we apply the Zero Product Property by setting each factor equal to zero: \( x = 0 \) and \( x - 4 = 0 \).

It is essential to understand the Zero Product Property because it allows us to conclude that for \( x(x - 4) = 0 \) to be true, either \( x = 0 \) or \( x - 4 = 0 \) must be true individually. Therefore, we can solve for 'x' by finding the values that make each factor equal to zero.

Square Roots Method

When a quadratic equation is in the form \( x^2 = a \) (where 'a' is a constant), we can use the Square Roots Method to find solutions. This method involves taking the square root of both sides of the equation to solve for 'x'. This process works because it essentially reverses the action of squaring 'x'.

However, it's crucial to remember that every positive number has two square roots: one positive and one negative. Therefore, when using the Square Roots Method, we must consider both potential solutions. For an equation like our example which has been simplified to \( x^2 = 16 \) (after factoring), the solutions would be \( x = \pm\sqrt{16} \) which simplifies to \( x = \pm4 \) because both 4 and -4, when squared, equal 16.