Problem 72 - Algebra for College Students

Question

Solve each of the following equations for \(x\). $$4 x^{2}+4 b x+b^{2}=0$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{-b}{2}\).

Step 1: Recognize the Equation Type

Notice that the given equation \(4x^2 + 4bx + b^2 = 0\) is a quadratic equation of the form \(ax^2 + bx + c = 0\) where \(a = 4\), \(b = 4b\), and \(c = b^2\).

Step 2: Check for Factoring

Observe that the given equation resembles the perfect square formula \((ax + b)^2 = a^2x + 2abx + b^2\). In this case, the equation can be written as \((2x + b)^2 = 0\).

Step 3: Solve the Factored Equation

Since the equation is now \((2x + b)^2 = 0\), apply the square root property, which implies \(2x + b = 0\).

Step 4: Solve for x

To solve \(2x + b = 0\), isolate \(x\) by subtracting \(b\) from both sides, resulting in \(2x = -b\). Then divide both sides by 2 to solve for \(x\): \[ x = \frac{-b}{2} \]

Key Concepts

Factoring Quadratics

Factoring quadratics is an algebraic method used to express a quadratic equation in the form \( ax^2 + bx + c = 0 \) as a product of two binomials. It simplifies the process of solving these equations by making it easier to apply further strategies such as the Zero Product Property. To factor a quadratic successfully, one must identify patterns or use techniques like grouping.

• The equation should ideally resemble a recognizable pattern, such as a simple trinomial or a perfect square trinomial.
• By factoring, complex expressions can be broken down, often simplifying the solution process significantly.
• Working through examples is the best way to become comfortable with various factoring methods.

In the given solution, our equation \( 4x^2 + 4bx + b^2 = 0 \) was factored once we recognized it follows the pattern of a perfect square trinomial, specifically the identity \((2x + b)^2 = 0\). This revelation allowed us to rewrite the equation into a more manageable form.

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic equation designed to be the square of a binomial. It can be expressed in the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). When expanded, it makes the process of factorizing much more straightforward, as the equation is already set up to form a perfect square.

• Identifying a perfect square trinomial involves confirming that both ends of the quadratic expression (the first and last terms) are perfect squares.
• The middle term should be twice the product of the square roots of the two perfect squares at the ends.
• This pattern simplifies the factorization and hastens the problem-solving process.

For our exercise, recognizing \( 4x^2 + 4bx + b^2 = (2x + b)^2 \) efficiently harnessed the concept of a perfect square trinomial, guiding the steps to solve the quadratic.

Solving Equations

Once you have successfully factored a quadratic equation or identified it as a perfect square trinomial, solving it often becomes simple. The key principle is to use techniques such as the Zero Product Property or the square root principle, to isolate the variable and find its value.

• With the equation \((2x + b)^2 = 0\), applying the square root principle helps simplify how we view it, essentially reducing it to \((2x + b) = 0\).
• Solving \(2x + b = 0\) means isolating \(x\) by standard algebraic manipulation, which often involves rearranging terms and dividing.
• Remember that solving these equations often involves basic arithmetic operations like addition, subtraction, multiplication, or division.

The simple solution here, \( x = \frac{-b}{2} \), illustrates how employing these methods efficiently brings clarity to quadratic problems.