Problem 30 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$x^{2}+7 x+2=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{7}{2} \pm \frac{\sqrt{41}}{2}\).

Step 1: Identify the quadratic equation

The given quadratic equation to solve is \(x^2 + 7x + 2 = 0\).

Step 2: Move the constant term to the right side

To begin completing the square, we rearrange the equation such that the quadratic and linear terms are on one side and the constant term is on the other: \(x^2 + 7x = -2\).

Step 3: Find the value to complete the square

Take the coefficient of \(x\), divide it by 2, and square it. This value will complete the square. Here, \((7/2)^2 = 49/4\).

Step 4: Add and subtract the perfect square

Add and subtract \(49/4\) to the left side of the equation to complete the square:\[x^2 + 7x + \frac{49}{4} - \frac{49}{4} = -2\]Simplifying, we have:\[x^2 + 7x + \frac{49}{4} = -2 + \frac{49}{4}\]

Step 5: Simplify and express as a binomial square

Simplify the right side of the equation:\(-2 + \frac{49}{4}\) becomes \(\frac{8}{4} + \frac{49}{4} = \frac{41}{4}\).Thus, the equation becomes:\[x^2 + 7x + \frac{49}{4} = \frac{41}{4}\]which can be rewritten as:\[(x + \frac{7}{2})^2 = \frac{41}{4}\]

Step 6: Solve for x by taking the square root on both sides

To solve for \(x\), take the square root of both sides:\[x + \frac{7}{2} = \pm \sqrt{\frac{41}{4}}\]Simplifying gives:\[x + \frac{7}{2} = \pm \frac{\sqrt{41}}{2}\]

Step 7: Isolate x

Subtract \(\frac{7}{2}\) from both sides to find \(x\):\[x = -\frac{7}{2} \pm \frac{\sqrt{41}}{2}\]Thus, the solutions are:\(x = -\frac{7}{2} + \frac{\sqrt{41}}{2}\) and \(x = -\frac{7}{2} - \frac{\sqrt{41}}{2}\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in mathematics and are often encountered in various fields such as physics, engineering, and economics. A quadratic equation is typically written in the standard form:

• \(ax^2 + bx + c = 0\)

where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). It is a second-degree polynomial equation as the highest power of the variable \(x\) is 2. Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the value of the discriminant \(b^2 - 4ac\).
Recognizing the properties of quadratic equations allows students to predict the nature of the roots, which is crucial for graphing parabolas and solving practical problems.

Solving Techniques: Completing the Square

Completing the square is a powerful technique for solving quadratic equations. It provides a systematic way to solve any quadratic equation, even allowing the quadratic formula to be derived. The method involves transforming a quadratic expression into a perfect square trinomial that can easily be solved.
Here are the steps to complete the square:

• Start with the quadratic equation in the form \(ax^2+bx+c=0\).
• Move the constant term \(c\) to the right side of the equation.
• Divide all terms by \(a\) (if \(a eq 1\)) to simplify.
• Take the coefficient of \(x\), divide it by 2, and square it; add and subtract this value on the left side to form a trinomial square.
• Rearrange the equation to express it as \((x + d)^2 = e\) where \(d\) is derived from the square's midpoint and \(e\) is the constant on the right.
• Solve for \(x\) by taking the square root of both sides and isolating \(x\).

Completing the square is particularly useful for understanding the derivation of the quadratic formula and solving non-factorable quadratics.

What are Binomial Squares?

A binomial square is the result of squaring a binomial expression, such as \((x + n)^2\). These squares expand to a trinomial, following the formula:

• \((x + n)^2 = x^2 + 2nx + n^2\)

This is an essential component of the completing the square method. When completing the square, we cleverly manipulate terms to form a perfect square trinomial. For example, in the quadratic equation \(x^2 + 7x + c\), to make a perfect square, we looked for the expression \((x + \frac{7}{2})^2 = x^2 + 7x + \frac{49}{4}\).
Creating binomial squares shows the hidden structure in quadratic expressions, making them easier to solve. Understanding this concept helps simplify complex algebraic expressions, making the process of solving quadratic equations more intuitive.