Problem 63 - Algebra for College Students

Question

Give a step-by-step description of how to solve \(3 x^{2}+9 x-\) \(4=0\) by completing the square.

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{3}{2} \pm \frac{\sqrt{129}}{6}\).

Step 1: Identify the Quadratic Equation

The equation given is a quadratic equation in the form \( ax^2 + bx + c = 0 \). In this problem, the equation is \(3x^2 + 9x - 4 = 0\). Here \(a = 3\), \(b = 9\), and \(c = -4\).

Step 2: Divide All Terms by 'a'

To complete the square, we start by making the coefficient of \(x^2\) equal to 1. We divide every term by the coefficient \(a = 3\). This gives us \(x^2 + 3x - \frac{4}{3} = 0\).

Step 3: Move the Constant Term to the Other Side

To complete the square, we need the expression \(x^2 + 3x\) on one side of the equation. Thus, add \(\frac{4}{3}\) to both sides: \(x^2 + 3x = \frac{4}{3}\).

Step 4: Find and Add the Square-Creating Value

Take the coefficient of \(x\), which is 3, divide by 2 to get \(\frac{3}{2}\), and square it to obtain \(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\). Add this value to both sides to form a perfect square trinomial: \(x^2 + 3x + \frac{9}{4} = \frac{4}{3} + \frac{9}{4}\).

Step 5: Simplify the Equation

The left side is now a perfect square: \((x + \frac{3}{2})^2\). On the right side, convert \(\frac{4}{3}\) into \(\frac{16}{12}\) and \(\frac{9}{4}\) into \(\frac{27}{12}\), then add them to get \(\frac{43}{12}\). This gives us \((x + \frac{3}{2})^2 = \frac{43}{12}\).

Step 6: Solve for 'x' by Taking Square Roots

Take the square root of both sides, remembering to consider both the positive and negative roots: \(x + \frac{3}{2} = \pm \sqrt{\frac{43}{12}}\).

Step 7: Isolate 'x'

Subtract \(\frac{3}{2}\) from both sides to solve for \(x\): \(x = -\frac{3}{2} \pm \sqrt{\frac{43}{12}}\).

Step 8: Simplify the Square Root Expression

Optionally, simplify \(\sqrt{\frac{43}{12}} = \sqrt{\frac{43}{12}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{129}}{6}\). Therefore the solutions are \(x = -\frac{3}{2} + \frac{\sqrt{129}}{6}\) and \(x = -\frac{3}{2} - \frac{\sqrt{129}}{6}\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and appear in the form \(ax^2 + bx + c = 0\). They are called "quadratic" because they involve the square of a variable (\(x^2\)). These equations can have two solutions because a parabola, which a quadratic equation graphically represents, can intersect a horizontal line at two points. Quadratic equations arise in various contexts such as calculating areas, physics problems involving motion, and maximizing or minimizing certain quantities. The standard form of a quadratic equation helps us apply different methods like factoring, using the quadratic formula, or completing the square to find the solutions. Understanding the structure of these equations is the first step toward solving them effectively.

Solving Quadratic Equations

When it comes to solving quadratic equations, there are several methods available, each suitable for different types of equations and factors like coefficients.

• Factoring: This method is effective when the equation can be rewritten as a product of two binomials. It involves finding two numbers that multiply to the constant term \(c\) and add up to the coefficient \(b\).
• Quadratic Formula: Useful for any quadratic equation, the formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) derives solutions directly and is handy for equations that are difficult to factor.
• Completing the Square: Ideal when rearranging the equation into a perfect square trinomial, as shown in this exercise, converting the equation into the format \((x + d)^2 = e\). This method transforms the equation, making it easier to solve by taking the square root of both sides.

Choosing the right method often depends on the characteristics of the quadratic equation. Each approach enhances understanding of the equation's roots and the nature of the solutions.

Algebraic Methods

In algebra, methods like completing the square are used to manipulate equations into forms that are easier to solve. The "completing the square" technique is pivotal for both solving quadratic equations and understanding their geometric representations.To complete the square, follow these key steps:

• Normalize the Quadratic Term: Ensure the leading coefficient of \(x^2\) is 1 by dividing the entire equation by \(a\) if necessary.
• Rearrange the Equation: Move the constant term \(c\) to the equation's other side, setting up an equation with \(x^2\) and \(x\) terms on one side.
• Add the Square Value: Calculate \(\left(\frac{b}{2}\right)^2\) and add it to both sides to form a perfect square on one side of the equation.
• Factor the Perfect Square: The expression becomes \((x + \frac{b}{2})^2 = \text{constant}\), allowing you to solve the equation by taking square roots.

These systematic steps help solve quadratic equations methodically. They offer not only a solution but also deep insight into the functionality and behavior of quadratic equations in both algebraic and geometric contexts.