Problem 63 - College Algebra

Question

Solve each equation in Exercises \(47-64\) by completing the square. $$3 x^{2}-2 x-2=0$$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = \frac{1}{3} + \sqrt{\frac{7}{9}}\) and \(x = \frac{1}{3} - \sqrt{\frac{7}{9}}\).

Step 1: Rearrange the equation

Rewrite the given equation in form \(ax^{2} + bx = -c\), so the equation becomes: \(3x^2 - 2x = 2\)

Step 2: Make the coefficient of \(x^{2}\) equal to one

Divide the whole equation by 3: \(\frac{3x^2}{3} - \frac{2x}{3} = \frac{2}{3}\) Simplifying, the equation is: \(x^2 - \frac{2}{3}x = \frac{2}{3}\)

Step 3: Complete the square

The next step is to complete the square on the left side of the equation. We should add \((b/2a)^2 = \left(\frac{2}{6}\right)^2 = \frac{1}{9}\) to both sides of the equation in order to make the left side of the equation a perfect square. Now the equation is: \(x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{2}{3} + \frac{1}{9}\)

Step 4: Simplify both sides of the equation

The left side will be simplified into the form: \((x - b/2a)^2\), and the right side will be a sum of \(\frac{2}{3}\) and \(\frac{1}{9}\). So the equation is: \((x - \frac{1}{3})^2 = \frac{2}{3} + \frac{1}{9} = \frac{7}{9}\)

Step 5: Solve for x

Now take square root of both sides of the equation: \(x - \frac{1}{3} = \sqrt{\frac{7}{9}}\), and \(x - \frac{1}{3} = -\sqrt{\frac{7}{9}}\). Solving for x gives: \(x = \frac{1}{3} + \sqrt{\frac{7}{9}}\) and \(x = \frac{1}{3} - \sqrt{\frac{7}{9}}\)

Key Concepts

Quadratic Equations

In mathematics, quadratic equations are a type of polynomial equation where the highest degree of the variable is two. They generally have the standard form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \, eq \, 0\).

Quadratic equations can be found in various real-life applications, such as calculating areas, determining the motion of objects, and solving problems that involve certain financial calculations. Working with quadratics can involve different methods such as factoring, using the quadratic formula, or completing the square.

The method of completing the square is particularly useful to understand the vertex form of a quadratic function, helping to identify its features like the vertex and axis of symmetry more easily. By rewriting the equation in a form that creates a perfect square trinomial, the solution to the quadratic equation becomes more straightforward.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Understanding algebraic expressions involves knowing how to manipulate these parts to form equations or find solutions.

• **Constants:** Numbers on their own, such as 2 or -1
• **Variables:** Symbols that represent unknown numbers, typically \(x\) or \(y\)
• **Coefficients:** Numbers that multiply the variables, like the 3 in \(3x^2\)

In our quadratic equation \(3x^2 - 2x - 2 = 0\), rearranging and factoring are key skills for working with algebraic expressions. When we complete the square for this equation, we alter the way the expression appears, helping us to solve for \(x\).

By understanding different forms of expressions and how to convert them, we gain better flexibility in solving equations, hence achieving accurate results efficiently.

Solving Equations

Solving equations involves finding the value(s) of variables that make the equation true. Here, for quadratic equations, one effective method is completing the square.

**Steps in Completing the Square:**

• Rearrange the equation to isolate the terms with variables on one side.
• Divide all terms by the coefficient of \(x^2\) if it's not 1.
• Add a constant to both sides to form a perfect square trinomial on one side.
• Rewrite the trinomial as a squared binomial.
• Solve for \(x\) by taking the square root of both sides and isolating \(x\).

For example, in the equation \(3x^2 - 2x - 2 = 0\), we first rearrange and divide to transform the expression into \((x - \frac{1}{3})^2 = \frac{7}{9}\).

Solving this gives the solution \(x = \frac{1}{3} \pm \sqrt{\frac{7}{9}}\), showing the potential roots of the equation. Completing the square effectively simplifies the problem, making it easier to understand and solve.