Problem 15 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$x^{2}+4 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \( x = -2 + \sqrt{6} \) and \( x = -2 - \sqrt{6} \).

Step 1: Move the constant to the other side

Start with the equation \( x^2 + 4x - 2 = 0 \). Move the constant term (-2) to the other side of the equation: \( x^2 + 4x = 2 \).

Step 2: Complete the square

To complete the square, take half of the coefficient of \( x \), which is 4. Half of 4 is 2, and then square it to get 4. Add this value to both sides of the equation: \( x^2 + 4x + 4 = 2 + 4 \).

Step 3: Write in perfect square form

The left side of the equation \( x^2 + 4x + 4 \) is now a perfect square trinomial. It can be written as \( (x+2)^2 \). Thus the equation becomes \( (x+2)^2 = 6 \).

Step 4: Solve for x

Take the square root of both sides: \( x+2 = \pm \sqrt{6} \). Solve for \( x \) by subtracting 2 from both sides: \( x = -2 \pm \sqrt{6} \).

Step 5: Write the solutions

The solutions to the equation are \( x = -2 + \sqrt{6} \) and \( x = -2 - \sqrt{6} \).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a eq 0 \). Each term represents part of a parabola when graphed. Here, \( x \) stands for the variable, while \( a \), \( b \), and \( c \) are known as coefficients and constants. The goal is to find the values of \( x \) that satisfy the equation, known as roots or solutions.

• The highest power of \( x \) is 2, making it a quadratic equation.
• The graph of a quadratic equation forms a parabola.

To solve these equations, there are several methods available:

• Factoring: If the quadratic can be factored into simpler binomials.
• Quadratic Formula: A formula that provides the solution \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This is applicable to all quadratic equations.
• Completing the Square: A technique used to transform the quadratic into a perfect square trinomial.

Understanding quadratic equations is crucial because they appear in numerous mathematical applications.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form \((x + d)^2\), where \( d \) is a real number. For example, \( x^2 + 4x + 4 \) is a perfect square trinomial because it can be expressed as \((x + 2)^2\).

• When completing the square, we aim to convert a standard quadratic into a perfect square trinomial.
• This process involves adjusting the constant term to help rewrite the quadratic expression.

Completing the square is typically used when the quadratic equation does not neatly factor. Here's the step-by-step process:

• Isolate the quadratic and linear terms.
• Take half of the linear coefficient, square it, and add it to both sides of the equation.
• The expression on one side of the equation forms a perfect square, which is then simplified into a binomial.

By creating a perfect square trinomial, solving the equation becomes straightforward.

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that satisfy the equation. For quadratic equations, one can use the method of completing the square among others. Here’s a quick breakdown:

• Move the constant term to the opposite side of the equation to set up for completing the square.
• Modify the equation so that the left-hand side becomes a perfect square trinomial.
• Rewrite the left side as a binomial square, equaling it to the adjusted constant on the right.
• Take the square root of both sides to solve for the variable.
• Since squaring can give both positive and negative results, consider both when solving. This is why solutions are often presented as \(x = a \pm b\).

The real benefit of mastering equation-solving methods like completing the square is that they allow us to analyze more complicated mathematical problems that rely on quadratic expressions.