Problem 61 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ z^{2}+3 z-4=0 $$

Step-by-Step Solution

Answer

The solutions are \(z = 1\) and \(z = -4\).

Step 1: Move the constant term

Start with the given equation: \[ z^{2} + 3z - 4 = 0 \]To complete the square, first move the constant term to the other side of the equation:\[ z^{2} + 3z = 4 \]

Step 2: Find the term to complete the square

Take half of the coefficient of the linear term (\(3\)), square it, and add it to both sides of the equation. The coefficient of \(z\) is \(3\), so half of it is \(\frac{3}{2}\), and its square is \[ \left(\frac{3}{2}\right)^{2} = \frac{9}{4} \]Add \(\frac{9}{4}\) to both sides:\[ z^{2} + 3z + \frac{9}{4} = 4 + \frac{9}{4} \]

Step 3: Write the left side as a square

Now the left side of the equation is a perfect square trinomial. Thus, it can be written as the square of a binomial:\[ (z + \frac{3}{2})^2 = \frac{25}{4} \]

Step 4: Solve for z by taking the square root

Take the square root of both sides of the equation:\[ z + \frac{3}{2} = \pm \sqrt{\frac{25}{4}} \]Calculating the square root gives:\[ z + \frac{3}{2} = \pm \frac{5}{2} \]

Step 5: Solve for z

Subtract \(\frac{3}{2}\) from both sides to find the values of \(z\):\[ z = -\frac{3}{2} + \frac{5}{2} \quad \text{or} \quad z = -\frac{3}{2} - \frac{5}{2} \]These simplify to:\[ z = 1 \quad \text{or} \quad z = -4 \]

Key Concepts

Quadratic Equations

Quadratic equations form the backbone of a vast array of mathematical problems and solutions. These types of equations are in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The highest power of the variable is 2, which gives the equation its name: 'quadratic', meaning square. Why are they important? Well, they describe parabolic motions such as the path of a thrown ball and can even be used in business to calculate profit and loss.

Solving quadratic equations can be approached in various ways, such as factoring, using the quadratic formula, or completing the square. Each method has unique advantages based on the specific problem and context. Understanding the nature of quadratic equations will allow you to choose the best method to find solutions efficiently.

Perfect Square Trinomial

A perfect square trinomial is a special form of a polynomial. This happens when a binomial is squared, resulting in an expression such as \( (x + d)^2 = x^2 + 2dx + d^2 \). Recognizing and creating perfect square trinomials is essential when completing the square.

• The process involves identifying the middle term, halving it, and then squaring the result.
• For example, with \( z^2 + 3z \), half of 3 is \( \frac{3}{2} \), squaring it gives \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \).

By adding \( \frac{9}{4} \) to both sides, we transform our equation into a perfect square trinomial: \( z^2 + 3z + \frac{9}{4} = 4 + \frac{9}{4} \). Knowing how to create and utilize perfect square trinomials is a powerful algebraic tool for simplifying equations and solving them.

Solving Equations

Solving equations is about finding the values of variables that make the equation true.

• A systematic approach is beneficial when working with any equation.
• In the context of quadratics, start by rearranging substances such as terms, so we isolate the variable in a stepwise manner.

For the equation \( z^2 + 3z - 4 = 0 \), moving the constant (-4) gives us \( z^2 + 3z = 4 \). At this point, we use techniques like completing the square to transform our equation into a form that allows us to easily deduce the values of \( z \). Completing the square leads us to a configuration where the problem simplifies significantly, making it more approachable.

Square Root Method

The square root method is a strategic technique for solving equations, especially after completing the square. Once you have a perfect square trinomial, it can be turned into an equation of the form \( (x + ")^2 = n \). To find \( x \), simply take the square root of both sides.

• For example, from \( (z + \frac{3}{2})^2 = \frac{25}{4} \), apply the square root method to both sides.
• This yields \( z + \frac{3}{2} = \pm \frac{5}{2} \).

Taking the square root introduces the consideration of both positive and negative roots. This is because both numbers squared will give the same positive square. Finally, solve for the variable by performing regular algebraic operations, in this case subtracting \( \frac{3}{2} \) to find \( z = 1 \) or \( z = -4 \). The square root method efficiently narrows down possible solutions, making solving equations smoother and more intuitive.