Problem 33 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by completing the square. $$x^{2}+4 x=-3$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = -4\).

Step 1: Reorganize The Equation

To start, the equation should be arranged with x-terms on one side and the constant on the other. So, the equation becomes \(x^{2} + 4x + 3 = 0\)

Step 2: Complete The Square

In order to complete the square, we need to add a value to both sides of the equation that makes the left side a perfect square trinomial. This value is \((b/2)^{2}\), here \(b\) is the coefficient of \(x\). In our case \(b = 4\), so the value we need to add is \((4/2)^{2} = 4\), which gives us a new equation: \(x^{2} + 4x + 4 = 0 + 4\) or simplified \(x^{2} + 4x + 4 = 4\)

Step 3: Rewrite Left Side as a Square

The left side of the equation is now a perfect square trinomial. It can be written as \( (x + 2)^{2}\). Therefore, the equation now reads: \((x + 2)^{2} = 4\)

Step 4: Solve for 'x'

Take the square root of both sides of the equation. This gives \(x + 2 = \pm 2\). This results in two solutions for 'x': \(x = 2 - 2 = 0\) and \(x = -2 - 2 = -4\).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations, making it a powerful tool in algebra.
This technique transforms a quadratic equation into a perfect square trinomial, which is an expression that can be easily factored as a square of a binomial.
Here's how it works:

• First, ensure that the quadratic equation is in the standard form of \[ax^2 + bx + c = 0\].
• Move any constant terms to the opposite side of the equation.
• Find the term that will complete the square: Take half of the coefficient of \(x\), square it, and add this square to both sides of the equation.
• This process turns the quadratic expression on one side of the equation into a perfect square trinomial.

By completing the square, you simplify the equation into a simple expression that reveals the roots directly, making it straightforward to solve.

Perfect Square Trinomial

A perfect square trinomial is a special kind of quadratic expression that can be expressed as the square of a binomial.
It takes the form of \((a+b)^2 = a^2 + 2ab + b^2\).
In the context of solving quadratic equations, once the left side of the equation is a perfect square trinomial, it becomes easier to work with.
Consider \(x^2 + 4x + 4\), which can be rewritten as \((x + 2)^2\).
This transformation outlines the structure and helps in simplifying the equation.

• Identify the quadratic term, \(x^2\), the linear term, \(bx\), and use them to find the value \((b/2)^2\) to complete the square.
• Rewriting the trinomial as a square of a binomial confirms that it is a perfect square trinomial.

Having the perfect square trinomial means you can directly equate it and solve for \(x\), which simplifies the process immensely.

Solving Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation, meaning the solutions or roots.
After completing the square and reaching a perfect square trinomial, solving becomes straightforward.
The following steps outline the process:

• Rewrite the trinomial in the form of \((x + n)^2 = k\), where \(n\) and \(k\) are constants.
• Take the square root of both sides, ensuring you consider both the positive and negative roots: \(x + n = \pm \sqrt{k}\).
• Solve for \(x\) by isolating it on one side of the equation, which typically gives two solutions.

Using these steps guarantees that you find all possible solutions to the quadratic equation, demonstrating a clear path from completing the square to arriving at the final solutions for \(x\).