Problem 45 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ x^{2}+8 x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -4 + \sqrt{15} \) and \( x = -4 - \sqrt{15} \).

Step 1: Move the Constant Term

To complete the square, first move the constant term to the right side of the equation. This gives\[ x^2 + 8x = -1 \].

Step 2: Prepare to Complete the Square

To complete the square, we need to find the number that makes the expression \( x^2 + 8x + ? \) a perfect square trinomial. Take half of the coefficient of \( x \), which is 8, so you get \( \frac{8}{2} = 4 \). Then square it to get 16.

Step 3: Add and Subtract the Square

Add and subtract 16 from the left side of the equation to complete the square: \[ x^2 + 8x + 16 - 16 = -1 \].

Step 4: Simplify the Equation

Rewrite the left side as a perfect square and simplify: \[ (x + 4)^2 - 16 = -1 \]. Move the -16 to the right: \[ (x + 4)^2 = 15 \].

Step 5: Solve for x

Take the square root of both sides to solve for \( x \). Remember to consider both positive and negative roots: \[ x + 4 = \pm \sqrt{15} \].

Step 6: Isolate x

Subtract 4 from each side: \[ x = -4 \pm \sqrt{15} \].

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. These equations result in a parabola when graphed on a coordinate plane. The solutions, or roots, of a quadratic equation can be found using several methods, such as factoring, using the quadratic formula, or completing the square, which we'll discuss here.

When solving quadratic equations by completing the square, our goal is to transform the equation into a perfect square trinomial, which allows us to take the square root of both sides easily. This method is particularly useful when the equation cannot be easily factored.

• Quadratic equations can have two solutions, a double solution (when the discriminant is zero), or no real solution if the solutions are complex numbers.
• They are widely used in various fields such as physics, engineering, and finance to model problems and calculate outcomes.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression of the form \((x + p)^2 = x^2 + 2px + p^2\). It results from squaring a binomial expression. Recognizing and forming a perfect square trinomial is a crucial step in completing the square. By transforming a quadratic expression into this form, we simplify the process of solving for the variable.

To create a perfect square trinomial from a quadratic expression like \(x^2 + bx\), one must follow these steps:

• Take half of the coefficient of the linear term (\(b\)). In our example, it would be \(\frac{b}{2}\).
• Square this value to get \(\left(\frac{b}{2}\right)^2\). This will be the term added and subtracted to create a perfect square trinomial.

Using this approach, you can rewrite \(x^2 + 8x + 1\) to a manageable expression \((x + 4)^2\) minus the extra terms, simplifying solving the equation substantially.

Square Root Method

The square root method is a straightforward way to solve equations once they have been expressed as a perfect square trinomial. Once the quadratic equation is set in the form \((x + p)^2 = d\), we can easily "undo" the square by taking the square root of both sides.

Here's how you can use the square root method:

• Take the square root of both sides of the equation \((x + p)^2 = d\) to get \(x + p = \pm \sqrt{d}\).
• Remember to consider both the positive and negative square roots, which correspond to the two potential solutions.
• Isolate \(x\) by performing arithmetic operations, such as adding or subtracting, to solve for the variable.

This method capitalizes on the properties of exponents and roots, helping solve what otherwise might be intricate algebraic equations with ease.