Problem 10 - Algebra 1

Question

Solve by completing the square. $$x^{2}-2 x-18=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation by completing the square are \(x = 1 + \sqrt{19}\) and \(x = 1 - \sqrt{19}\).

Step 1: Rewrite quadratic equation

Let's rewrite the quadratic equation such that we isolate the variable terms on one side. We get the quadratic equation in the form \(x^{2} - 2x = 18\).

Step 2: This part of the process involves 'completing the square' with the equation

To complete the square, we should add \((-b/2)^2\) to both sides of the equation, where \(b = -2\) is the coefficient of \(x\). So we add \((-(-2)/2)^2 = 1\) to both sides to get \(x^{2} - 2x + 1 = 18 + 1\). Simplifying the right hand side we get \(x^{2} - 2x + 1 = 19\).

Step 3: Rewrite the equation as a squared binomial

Now, left hand side of the equation can be rewritten as a perfect square. Specifically it is \((x - 1)^2\) because \((a - b)^{2} = a^{2} - 2ab + b^{2}\). So we rewrite the equation as \((x - 1)^2 = 19\).

Step 4: Solve for x

We solve for \(x\) by taking square root on both sides. We get \(x - 1 = \pm \sqrt{19}\). Finally, we add 1 to both sides to isolate \(x\). Hence the solutions are \(x = 1 + \sqrt{19}\) and \(x = 1 - \sqrt{19}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The highest degree of a quadratic equation is two, meaning the variable \(x\) is squared.

Quadratic equations can be found in numerous applications such as physics for describing motion, in finance for calculating compound interest, and in engineering for designing parabolic structures. Recognizing a quadratic equation involves identifying the squared term, the linear term, and a constant:

• The \(x^2\) term is the quadratic term.
• The \(x\) term is the linear term.
• The constant \(c\) is the free term without a variable.

Understanding that these three components make up a quadratic equation helps in setting up the equation properly for solving.

Solving Quadratic Equations

There are several methods to solve quadratic equations, and completing the square is one of them. Each method offers a way to find the roots or solutions of the equation where it equals zero. The goal of solving is to determine the value(s) of \(x\) that satisfy the equation.

When it comes to completing the square, it involves changing the form of the quadratic equation to make it easier to solve. Here’s a quick overview of the strategy:

• Rearrange the equation to have the variable terms on one side and the constant on the other.
• Makes the equation into something easily solvable by converting it into a perfect square trinomial.
• Takes the square root of both sides to solve for the variable \(x\).

This process can be especially helpful when the equation doesn't easily factor and when using the quadratic formula seems complex or tedious.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. This occurs in the form \( (a + b)^2 = a^2 + 2ab + b^2 \). Recognizing this pattern is essential in simplifying quadratic equations through the method of completing the square.

To create a perfect square trinomial from a quadratic expression, you follow these steps:

• Take half of the coefficient of the \(x\) term (the linear term).
• Square this value.
• Add this squared number back to both sides of the equation.

This transforms the expression into a perfect square trinomial, making it solvable by expressing it as the square of a binomial. For example, in the equation \(x^2 - 2x = 18\), you add \(1\) (since \((-2/2)^2 = 1\)) to both sides to form \((x-1)^2 = 19\). Understanding how to manipulate expressions into perfect square trinomials enables more straightforward solutions for quadratic equations.