Problem 13 - Algebra 2

Question

Solve each equation by completing the square. \(3 x^{2}+12 x-18=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -2 + \sqrt{10}\) and \(x = -2 - \sqrt{10}\).

Step 1: Move constant to right side

First, rearrange the given equation so that the constant term is on the right side of the equation, we start with: \[3x^2 + 12x = 18\]

Step 2: Divide by the coefficient of x²

Next, divide every term by the coefficient of the quadratic term, which is 3. This will simplify the equation to have the quadratic coefficient as 1:\[x^2 + 4x = 6\]

Step 3: Identify the coefficient of x and complete the square

Take the coefficient of the linear term \(4\) (the coefficient in front of \(x\)), divide it by 2, and square it. This gives:\[ \left(\frac{4}{2}\right)^2 = 4 \]

Step 4: Add and subtract the squared number

Add and subtract 4 to keep the equation balanced. Insert this into the equation:\[x^2 + 4x + 4 - 4 = 6\]

Step 5: Rewrite as a perfect square

Now, the expression \(x^2 + 4x + 4\) can be written as the square of a binomial:\[(x + 2)^2 - 4 = 6\]

Step 6: Solve the equation for x

Add 4 to both sides to isolate the square term:\[(x + 2)^2 = 10\]Take the square root of both sides, remembering to consider both the positive and negative roots:\[x + 2 = \pm \sqrt{10}\]Finally, solve for \(x\) by subtracting 2 from both sides:\[x = -2 \pm \sqrt{10}\]

Step 7: Conclusion: Solution of the equation

The solutions of the equation \(3x^2 + 12x - 18 = 0\) by completing the square are:\[x = -2 + \sqrt{10}\] and \[x = -2 - \sqrt{10}\]

Key Concepts

Quadratic Equations

Quadratic equations are a foundational element of algebra and represent polynomial expressions of the form \( ax^2 + bx + c = 0 \). These equations are known for having a degree of 2, which means the highest exponent of the variable \( x \) is 2.

Understanding the structure of a quadratic equation involves recognizing three key components:

• The quadratic term \( ax^2 \), where \( a eq 0 \).
• The linear term \( bx \), which can affect the shape and position of the parabola.
• The constant term \( c \), which can shift the parabola up or down on the graph.

Quadratic equations are essential because they describe parabolic curves, and can appear in various scenarios such as projectile motion and statistics.

When solving quadratic equations, there are several methods available, including factoring, using the quadratic formula, and completing the square. Each method has its particular advantage depending on the form of the quadratic equation.

Perfect Square Trinomial

A perfect square trinomial is a special format of a quadratic expression that can be expressed as the square of a binomial.

This means an expression like \( (x + d)^2 \) leads to a trinomial \( x^2 + 2dx + d^2 \). This pattern is vital when completing the square, as it allows us to rewrite a quadratic in a more manageable form.

For example, if you have \( x^2 + 4x + 4 \), you can recognize it as the perfect square trinomial \( (x + 2)^2 \).

The development of this pattern comes from:

• Taking half of the linear coefficient \( b \) (in this case, \( 4/2 = 2 \))
• Squaring this result to complete the square and form the trinomial \( x^2 + 4x + 4 \)

Using perfect square trinomials greatly simplifies solving quadratic equations because it transforms the equation into a format that is easier to manipulate.

Solving Equations

Solving quadratic equations by completing the square involves transforming the standard quadratic equation into a perfect square trinomial. This method provides an effective way to find the solutions or roots of the equation.

Let's break down the process used in the solution:

• First, isolate the quadratic and linear terms by moving constants to one side.
• Make sure the coefficient of \( x^2 \) is 1 by dividing the entire equation if needed.
• Complete the square by adding and subtracting the same value, this forms a perfect square trinomial on one side.
• Once a trinomial is written as a binomial squared, solve the equation by taking the square root of both sides.
• Don't forget to consider both the positive and negative roots when taking the square root.

This approach not only solves the equation \( 3x^2 + 12x - 18 = 0 \) but also reinforces a strategic way to solve any quadratic equation analytically.

In the given problem, this results in the solutions \( x = -2 + \sqrt{10} \) and \( x = -2 - \sqrt{10} \). This demonstrates the power of completing the square in solving even more complex quadratic equations efficiently.