Problem 23 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}+3 x-1=0\)

Step-by-Step Solution

Answer

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

Step 1: Re-arranging the quadratic equation

Rearrange the equation so that only the constant term is on the right side. Doing this for the given equation gives:\(x^{2} + 3x = 1\)

Step 2: Completing the square on the left side

To complete the square on the left side, add square of half the coefficient of x to both sides, which makes the left side a perfect square. Half of the coefficient of x is \(3/2\). Squaring this gives \( (3/2)^2 = 2.25\). Adding this to both sides results in:\(x^{2} + 3x + 2.25 = 1 + 2.25\)Simplifying gives:\((x + 1.5)^2 = 3.25\)

Step 3: Solving for x

Now, we can take the square root on both sides of the equation and solve for x, so:\(x + 1.5 = ± \sqrt{3.25}\)Subtracting 1.5 from both sides gives values of x:\(x = -1.5 ± \sqrt{3.25}\)

Key Concepts

Completing the Square

The method of completing the square involves transforming a quadratic equation into a simple equation that has a perfect square trinomial on one side and a constant on the other. This technique is particularly useful when the quadratic equation does not factor easily or when you're working with equations where the coefficient of the quadratic term, typically denoted as 'a,' is 1.

To master this approach, follow these steps: first, ensure that the equation is in the standard quadratic form of ax^2 + bx + c = 0. If 'a' is not 1, divide the entire equation by 'a' to normalize the coefficient. Next, move the constant term 'c' to the other side of the equation by adding or subtracting it from both sides. Now, you're ready to create the perfect square.

Finding the Perfect Square

To complete the square, take half of the 'b' coefficient, square it, and add it to both sides of the equation. For example, if your equation after normalization is x^2 + 3x = 1, you take half of 3 (which is 1.5), square it (to get 2.25), and then add it to both sides. You will get a new equation (x + 1.5)^2 = 3.25, where the left side is now a perfect square trinomial. The equation is easier to solve, as it involves finding the square root of both sides to find 'x'.

Algebraic Manipulation

Performing algebraic manipulation is a fundamental skill in solving algebra problems, particularly involving quadratic equations. It encompasses a series of operations like adding, subtracting, multiplying, dividing, and factoring, which are used to re-arrange and simplify algebraic expressions and equations.

In the context of completing the square, algebraic manipulation is used to isolate variable terms on one side and constants on the other, as well as to add the same value to both sides without changing the equation's balance. Understanding this skill is critical because a single error in manipulation can lead to incorrect results.

Practice Makes Perfect

To improve your confidence, practice by re-arranging and simplifying various forms of equations. This not only applies to quadratics but to all algebraic equations. It's essential when aiming to simplify more complex problems or when setting up equations for certain techniques like completing the square.

Perfect Square Trinomials

A perfect square trinomial is an expression formed by squaring a binomial. It's a special product that follows the pattern (ax + b)^2 = a^2x^2 + 2abx + b^2. Recognizing these patterns can greatly simplify solving quadratic equations, especially when you're completing the square.

The process of completing the square essentially aims to transform a given quadratic equation into a perfect square trinomial. When you have an equation such as x^2 + 3x + 2.25, it is indeed a perfect square trinomial because it can be written as (x + 1.5)^2, where 'x + 1.5' is the binomial and '2.25' is the square of '1.5'.

Identifying the Square

For your equation-solving arsenal, learning to quickly identify a perfect square trinomial is invaluable. It will enable you to work backward from the standard form of a quadratic to its factored form and thus find the solution faster and with greater accuracy.