Problem 87 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(4 x^{2}-16=0\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2\) and \(x = -2\)

Step 1: Factor out the Common Factor

First, factor out the common factor from each term, that is 4:\n\(4x^{2} - 16 = 0 \)\nwhen Factoring 4 from each term, the equation is then simplified to:\n\(4(x^{2} - 4) = 0\)

Step 2: Applying difference of squares

The expression \(x^{2} - 4\) is a difference of squares, which can be factored as \((x-2)(x+2)\). So the equation becomes:\n\(4(x - 2)(x + 2) = 0\)

Step 3: Equate Each Factor to Zero

Set each factor to zero and solve for \(x\):\n\(x - 2 = 0 \)\nto give \(x = 2\)\nand\n\(x + 2 = 0\)\nto give \(x = -2\)

Key Concepts

Factoring

Factoring is a method used to simplify expressions or solve equations by writing them as a product of their factors. It's like breaking down a number or expression into pieces that multiply together to give the original expression. For instance, when you see an equation like \(4x^2 - 16 = 0\), the first step is to look for any common factors in all terms. Here, both terms have a factor of 4.

Factoring out the 4, the equation becomes \(4(x^2 - 4) = 0\). When you factor, you’re essentially dividing each term by the common factor and placing it outside of parentheses. Factoring is a crucial step because it simplifies the equation, making the next steps easier to handle. It's important to factor completely until you can’t break down the equation any further.

Difference of Squares

The difference of squares is a specific type of factoring pattern that is very helpful when dealing with quadratic equations. A difference of squares expression looks like \(a^2 - b^2\). This pattern can be factored using the formula \((a - b)(a + b)\).

In our step-by-step solution, we encounter the expression \(x^2 - 4\), which fits the difference of squares pattern nicely. Here, \(a\) is \(x\) and \(b\) is 2, since \(4 = 2^2\). Thus, \(x^2 - 4\) can be rewritten as \((x - 2)(x + 2)\).

Recognizing and applying the difference of squares formula transforms the quadratic into a product of two binomials. This transformation is vital because solving for \(x\) becomes straightforward once we express the quadratic as a product of two simpler factors.

Solving Equations

Solving quadratic equations, especially by factoring, is an effective method to find the values of \(x\) that satisfy the equation. Once the equation is factored, as with \(4(x-2)(x+2)=0\), the next step is to use the zero-product property. This property states that if the product of two expressions is zero, then at least one of the expressions must be zero.

So, set each factor equal to zero:

• \(x - 2 = 0\)
• \(x + 2 = 0\)

Solving these gives the solutions \(x = 2\) and \(x = -2\).

This step is where the magic happens in solving equations by factoring. It allows us to find where the original quadratic touches or crosses the x-axis on a graph, essentially revealing the solution to the equation. The clearer you understand these steps, the easier it becomes to handle more complex quadratic equations efficiently.