Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-30: Solve by completing the square. (Note: See the discussion after Example 5 for help in solving Exercises 29 and 30 .) $$ x^{2}-8 x+11=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 4 + \sqrt{5} \) and \( x = 4 - \sqrt{5} \).

Step 1: Identify the Coefficient

Identify the coefficient of the linear term and the constant term from the equation. The given equation is \( x^2 - 8x + 11 = 0 \). The linear coefficient \( b \) is \(-8\), and the constant term is \( 11 \).

Step 2: Move the Constant Term

Move the constant term to the other side of the equation to focus on the expression that needs to be completed: \( x^2 - 8x = -11 \).

Step 3: Find the Value to Complete the Square

To complete the square, use the formula \( \left( \frac{b}{2} \right)^2 \) where \( b \) is the coefficient of \( x \). Here, \( b = -8 \), so \( \left( \frac{-8}{2} \right)^2 = 16 \).

Step 4: Add the Square to Both Sides

Add \( 16 \) to both sides of the equation to complete the square: \( x^2 - 8x + 16 = 5 \).

Step 5: Write the Perfect Square Equation

The expression \( x^2 - 8x + 16 \) can now be written as a perfect square: \( (x - 4)^2 = 5 \).

Step 6: Solve the Perfect Square Equation

Take the square root of both sides to solve for \( x \): \( x - 4 = \pm \sqrt{5} \).

Step 7: Isolate the Variable

Solve for \( x \) by adding \( 4 \) to both sides: \( x = 4 \pm \sqrt{5} \). Therefore, the solutions are \( x = 4 + \sqrt{5} \) and \( x = 4 - \sqrt{5} \).

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 \( a eq 0 \). These equations are fundamental in algebra and appear frequently across various fields of mathematics and science. The characteristic feature of quadratic equations is the degree, which is 2, indicated by the highest power of \( x \) being 2. The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient \( a \).
The general solution to a quadratic equation can be found using methods like:

• Factoring, when it readily factors into two binomials
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square, which is useful for understanding the derivation of the quadratic formula and when the equation doesn’t factor easily

By understanding and applying these methods, the roots or solutions of the quadratic equation can be determined. Completing the square, in particular, is a valuable technique not just for solving but for graphing quadratics by expressing them in vertex form.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. With quadratic equations, this process requires a few systematic steps.
One effective method to solve quadratic equations is to complete the square, which is especially handy when the equation does not factor neatly. It involves several key steps:

• Isolate the quadratic and linear terms from the constant on one side to prepare for completing the square.
• Determine the term needed to make the quadratic expression a perfect square. Use \( \left( \frac{b}{2} \right)^2 \), where \( b \) is the coefficient of the linear term.
• Add this new term to both sides of the equation, thus maintaining the equation's balance.
• After rewriting the left side as a squared binomial, solve the equation by taking the square root of both sides.
• Finally, isolate the variable to find the solutions.

This step-by-step method simplifies the process, ensuring accuracy and a deep understanding of the structure of quadratic equations.

Algebraic Techniques

Algebraic techniques are essential tools in finding solutions to quadratic equations and in simplifying expressions. Completing the square is one such technique that transforms the equation into a more manageable form. Understanding this technique requires some basic algebraic manipulation and arithmetic skills.
Completing the square involves changing a quadratic expression into a perfect square trinomial, which can then be rewritten as the square of a binomial. This is achieved by:

• Identifying and isolating the quadratic term along with the linear term.
• Finding the specific value using \( \left( \frac{b}{2} \right)^2 \) to complete the square.
• Adding and subtracting this value to maintain the equation's integrity.
• Simplifying the equation so it can be solved easily using square roots.

Learning and applying algebraic techniques like completing the square is crucial in solving complex equations, enhancing problem-solving skills, and mastering algebra.