Problem 19 - Algebra and Trigonometry

Question

Solve the equation by completing the square. \(x^{2}+22 x+21=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -1\) and \(x = -21\).

Step 1: Move the Constant to the Right Side

To complete the square, we first move the constant term to the other side of the equation. Starting with the equation \( x^2 + 22x + 21 = 0 \), subtract 21 from both sides to obtain:\[ x^2 + 22x = -21 \]

Step 2: Find the Value to Complete the Square

To complete the square, take the coefficient of \(x\), which is 22, divide it by 2, and then square it. \[ \left(\frac{22}{2}\right)^2 = 11^2 = 121 \]Add this value to both sides of the equation to maintain equality.

Step 3: Add the Square to Both Sides

Add 121 to both sides of the equation:\[ x^2 + 22x + 121 = -21 + 121 \]Simplifying the right side gives:\[ x^2 + 22x + 121 = 100 \]

Step 4: Write the Left Side as a Perfect Square

The left side of the equation is now a perfect square trinomial and can be written as:\[ (x + 11)^2 \]Thus, the equation becomes:\[ (x + 11)^2 = 100 \]

Step 5: Solve by Taking the Square Root of Both Sides

Take the square root of both sides of the equation:\[ \sqrt{(x + 11)^2} = \pm \sqrt{100} \]Simplifying gives:\[ x + 11 = \pm 10 \]

Step 6: Solve for x

We have two solutions from the square root:1. \( x + 11 = 10 \)2. \( x + 11 = -10 \)For the first equation, \( x = 10 - 11 = -1 \).For the second equation, \( x = -10 - 11 = -21 \).Thus, the solutions are \( x = -1 \) and \( x = -21 \).

Key Concepts

Quadratic Equation

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). These equations are important because they can express diverse phenomena in fields like physics and engineering. Identifying a quadratic equation is straightforward as it includes a variable, usually \( x \), raised to the power of two.To solve quadratic equations, there are several methods available:

• Factoring - breaking it down into two simpler components.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square - converting it into a perfect square trinomial.

Choosing the right method often depends on the structure of the equation and the numbers involved. Completing the square is a powerful technique, particularly when other methods are cumbersome or not viable.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression where it can be written as the square of a binomial. This form is typically seen as \((x + d)^2\), which expands to \(x^2 + 2dx + d^2\). Recognizing and creating a perfect square trinomial involves looking at the middle term of a quadratic expression.
To transform \(x^2 + bx + c\) into a perfect square trinomial:

• Take the coefficient of \(x\), \( b \), divide it by 2, and square it. So it becomes \((\frac{b}{2})^2\).
• Add that squared value both to the left and right sides of the equation to maintain the equality.

For example, in the given equation \(x^2 + 22x + 21\), by adding \(121\) (calculated as \(11^2\)), the expression \(x^2 + 22x + 121\) forms the perfect square trinomial \((x + 11)^2\). This step is crucial in simplifying and solving the equation efficiently.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. When dealing with quadratic equations, completing the square is a particularly strategic method for deriving solutions. The process involves several key steps that simplify the quadratic into an equation that is easier to solve.
In completing the square, the constant gets moved initially:

• Start by equating the trinomial without a constant term, moving it to one side.
• Next, compute the necessary square addend so the expression on one side becomes a perfect square.

This reformulation allows the equation to translate into a binary form that unveils the square root method. Practically, solving continues with isolating terms, leading directly or indirectly to straightforward calculations.

Square Root Method

The square root method is the finalization phase of solving quadratic equations once they are written as a perfect square trinomial. Once the trinomial resembles the expression \((x+d)^2 = k\), extracting the root of both sides reveals the likely solutions.
When using the square root method:

• The equation produced — a binomial squared equals a number, say \( (x + 11)^2 = 100 \) — allows for the square root operation to simplify the expression.
• Take the square root of both sides, noting that the solution includes both the positive and negative roots, thus \( x + 11 = \pm 10 \).

These operations yield direct results to consider, in this example resulting in \( x = -1 \) and \( x = -21 \). This method emphasizes accuracy in calculation and perception of both potential answers.