Problem 47 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+3 x-1=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -1.5 + sqrt{3.25}\) and \(x = -1.5 - sqrt{3.25}\)

Step 1: Rewriting the Equation

First, the given equation \(x^2 + 3x - 1 = 0\) is rewritten as \(x^2 + 3x = 1\). This is simply accomplished by adding 1 to both sides of the equation.

Step 2: Completing the Square

The next step involves 'completing the square'. We can do this by taking half of the coefficient of x, squaring it and adding it to both sides. Here the coefficient of x is 3, so half of it is \(1.5 = 3/2\). Squaring it, we get \((3/2)^2 = 2.25\). So our equation is now \(x^2 + 3x + 2.25 = 1+2.25\) which simplifies to \(x^2 + 3x + 2.25 = 3.25\).

Step 3: Form a Perfect Square Trinomial

We can now rewrite the left side as a square of a binomial. The equation becomes \((x + 1.5)^2 = 3.25\).

Step 4: Solve for x

Applying the square root property to solve the equation, we find \(x + 1.5 = \pm sqrt{3.25}\). To get x, we subtract 1.5 from both sides which gives us \(x = -1.5 \pm sqrt{3.25}\)

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a central part of algebra that students encounter frequently. These equations take the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The equation represents a parabola when graphed on the Cartesian plane.

The solutions to quadratic equations can be found using several techniques, including factoring, using the quadratic formula, and completing the square.
All these methods aim to find the roots, or the x-values, where the equation equals zero.

• Factoring: Useful when the quadratic expression can be decomposed into a product of simpler expressions.
• Quadratic Formula: A reliable method that can always be applied to find the roots \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).
• Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.

In this article, we leverage the 'completing the square' method to solve the given quadratic equation.

Perfect Square Trinomial

A perfect square trinomial is a special quadratic form that can be expressed as the square of a binomial. It takes the form \((x + d)^2 = x^2 + 2dx + d^2\), where \(d\) is a constant. Recognizing this pattern allows us to simplify and solve quadratic equations with ease.

To convert a quadratic expression into a perfect square trinomial, you follow these steps:

• Identify the coefficient of the \(x\) term, which we'll call \( b \).
• Divide \( b \) by 2 and square the result. Add and subtract this square within the equation.
• You will have a perfect square trinomial: \( (x + \frac{b}{2})^2 \).

For example, in the equation \(x^2 + 3x\), half of 3 is \(1.5\) and squaring it gives us \((1.5)^2 = 2.25\). Adding and subtracting 2.25 allows us to form the trinomial: \( (x + 1.5)^2 \), simplifying the process of finding solutions.

Solving Equations Step by Step

Solving quadratic equations by completing the square involves a few methodical steps. Let's break it down using our example equation \(x^2 + 3x - 1 = 0\), which we aim to solve step by step.

**Step 1: Rearrange the Equation**

• Move the constant to the other side: \(x^2 + 3x = 1\)

**Step 2: Complete the Square**

• Take half of the \(3\), which is \(1.5\), and square it: \((1.5)^2 = 2.25\)
• Add this square to both sides: \(x^2 + 3x + 2.25 = 1 + 2.25\)
• Simplify to form the trinomial: \(x^2 + 3x + 2.25 = 3.25\)

**Step 3: Form a Perfect Square**

• Rewrite the left side as a squared binomial: \((x + 1.5)^2 = 3.25\)

**Step 4: Solve for \(x\)**

• Apply the square root property: \(x + 1.5 = \pm \sqrt{3.25}\)
• Isolate \(x\) by subtracting \(1.5\): \( x = -1.5 \pm \sqrt{3.25} \)

This stepwise method not only finds the solution but also builds a deeper understanding of how quadratic equations can be handled using algebraic principles.