Problem 44 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ x^{2}-7 x-1=0 $$

Step-by-Step Solution

Answer

\(x = \frac{7 + \sqrt{53}}{2}\) or \(x = \frac{7 - \sqrt{53}}{2}\).

Step 1: Move Constants

Start by moving the constant term to the other side of the equation:\[x^2 - 7x = 1\]

Step 2: Identify Coefficient and Complete the Square

To complete the square, take half of the coefficient of the linear term \(-7\), square it, and add it to both sides. Half of \(-7\) is \(-\frac{7}{2}\), and squaring it gives:\[\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\]Add this to both sides:\[x^2 - 7x + \frac{49}{4} = 1 + \frac{49}{4}\]

Step 3: Solve the Equation

The left side can now be written as a square:\[(x - \frac{7}{2})^2 = \frac{53}{4}\]Take the square root of both sides:\[x - \frac{7}{2} = \pm \sqrt{\frac{53}{4}}\]Which simplifies to:\[x - \frac{7}{2} = \pm \frac{\sqrt{53}}{2}\]

Step 4: Isolate x

Finally, solve for \(x\) by adding \(\frac{7}{2}\) to both sides:\[x = \frac{7}{2} \pm \frac{\sqrt{53}}{2}\]This results in two possible solutions for \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable with the highest exponent of the variable being 2. Generally, it is expressed in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not equal to zero.
This form is critical because it defines the quadratic nature of the equation. Quadratic equations have been studied for centuries and are fundamental due to their applications in various fields such as physics, engineering, finance, and statistics.
There are several methods to solve quadratic equations:

• Factoring: This works when a quadratic polynomial can easily be factored into the product of two binomials.
• Completing the square: This method involves rearranging the equation to form a perfect square trinomial, as demonstrated in the solution above.
• Quadratic formula: This is a formula derived from completing the square, useful for solving any quadratic equation: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Graphical method: Solving by finding the points where the parabola \( y = ax^2 + bx + c \) intersects the x-axis.

Each method provides a different perspective on understanding and solving quadratic equations. Depending on the equation, some methods may be more advantageous than others.

Solving Quadratic Equations

Solving quadratic equations can be approached through various techniques, and the choice often depends on the properties of the equation itself. Completing the square, as shown in the exercise, is one effective method.
This technique is particularly useful because it simplifies a quadratic equation into a form that allows for easy application of the square root principle, thus directly leading to the solution. Here's a quick breakdown of the process:

• First, rearrange the equation to set the constant to the other side, focusing the left-hand side on creating a perfect square.
• Determine half the coefficient of the linear term \( b \) (the coefficient of \( x \)), square it, and add it to both sides to transform the left side into a perfect square trinomial.
• Rewrite the equation as a perfect square and solve for \( x \) by taking the square root.

This approach not only facilitates solving the equation but also enhances understanding of the geometric properties inherent in quadratic expressions, bridging algebraic manipulation with geometric interpretation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition, subtraction, multiplication, and division). They are the building blocks of algebra and form the basis of equations. An equation is simply a statement that two algebraic expressions are equal.
In the context of quadratic equations, each part of the expression \( ax^2 + bx + c \) has a specific role:

• \( ax^2 \): The quadratic term, where \( a \) influences the parabola's width and direction of opening.
• \( bx \): The linear term, affecting the position and symmetry of the parabola.
• \( c \): The constant term, often determining the y-intercept.

Understanding these components is crucial because they help in visualizing the graph of the quadratic function and predicting its behavior.
By manipulating algebraic expressions, especially when completing the square, one can reveal hidden features of a quadratic equation, making the problem-solving process both precise and insightful.