Problem 16 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}+6 x+4=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -3 + \sqrt{5} \) and \( x = -3 - \sqrt{5} \).

Step 1: Move Constant to Right Side

We start with moving the constant term to the right side of the equation. Given equation: \[ x^2 + 6x + 4 = 0 \]Subtract 4 from both sides: \[ x^2 + 6x = -4 \]

Step 2: Complete the Square

To complete the square, add and subtract a new term that makes the left side a perfect square trinomial. The term to add and subtract is \((\frac{6}{2})^2 = 9\). Add 9 to both sides: \[ x^2 + 6x + 9 = -4 + 9 \]The equation becomes: \[ (x + 3)^2 = 5 \]

Step 3: Solve for x

Take the square root of both sides to get rid of the square: \[ x + 3 = \pm \sqrt{5} \]Subtract 3 from both sides to solve for \(x\): \[ x = -3 \pm \sqrt{5} \]

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This type of equation involves an unknown variable, \( x \), raised to the power of 2, making it different from linear equations which have the highest power of 1. Quadratic equations can have up to two distinct solutions, also known as roots. These roots can be real or complex numbers depending on the discriminant (\( b^2 - 4ac \)).

There are several methods to find the solutions of quadratic equations, including:

• Factoring the quadratic equation into two binomials.
• Using the quadratic formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).
• Completing the square, a useful method particularly when the quadratic does not factor easily.
• Graphical methods, plotting the curve to find intersection points with the x-axis.

Completing the square, used in the solution above, is a powerful technique to transform the equation into a perfect square trinomial, making it easier to solve.

Simplest Radical Form

Expressing an answer in simplest radical form means ensuring that any square roots within an equation are simplified such that there are no perfect square factors remaining inside the radical. It refers to simplifying the radical expression as much as possible while retaining the square root symbol. For example, \( \sqrt{50} \) can be simplified to \( 5\sqrt{2} \) because 50 can be divided into the product of 25 and 2, and \( \sqrt{25} \) is 5.

In the context of solving quadratic equations, once a solution is found, it is often presented in simplest radical form to provide the exact values rather than a decimal approximation.

• Begin by identifying any perfect square that may be a factor of the number inside the square root.
• Extract the square root of the perfect square, simplifying the radical.
• Ensure no further simplifications are possible, keeping the answer as concise as possible.

This process is helpful in ensuring that answers are both accurate and presented in a uniform standard across mathematical problems.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that is the square of a binomial. In mathematical terms, a trinomial \( ax^2 + bx + c \) is a perfect square trinomial if it can be expressed as \( (mx + n)^2 \), where \( m \) and \( n \) are real numbers. This means:

• The first term \( ax^2 \) is a perfect square.
• The last term \( c \) is a perfect square.
• The middle term \( bx \) is twice the product of the square roots of the first and last terms.

For example, consider the equation \( x^2 + 6x + 9 \). By completing the square, we transformed \( x^2 + 6x + 4 \) into a perfect square trinomial \( (x + 3)^2 \). Completing the square involves carefully selecting a term which, when added and then subtracted, completes the square, ensuring symmetry in the equation.

Understanding perfect square trinomials is key in simplifying quadratic equations, allowing easier manipulation and solving. This optimal approach enhances solving efficiency, especially when other methods like factoring are not straightforward.