Problem 19 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by completing the square. $$v(v+4)=6$$

Step-by-Step Solution

Answer

The solutions are \(v = \sqrt{10} - 2\) and \(v = -\sqrt{10} - 2\).

Step 1: Expand the Equation

First, we expand the left-hand side of the equation \(v(v+4)\) to get \(v^2 + 4v\). The equation becomes \(v^2 + 4v = 6\).

Step 2: Move Constant to the Right

Subtract 6 from both sides to move the constant to the right-hand side: \(v^2 + 4v - 6 = 0\).

Step 3: Prepare to Complete the Square

To complete the square, we need the coefficient of \(v^2\) to be 1, and here it is already 1. We then take half of the coefficient of \(v\), square it, and add it to both sides. The coefficient of \(v\) is 4, so \(\frac{4}{2} = 2\) and \(2^2 = 4\).

Step 4: Add and Subtract the Square

Add and subtract 4 within the equation on the left side: \(v^2 + 4v + 4 - 4 = 6\). Rewriting gives us \((v^2 + 4v + 4) - 4 = 6\).

Step 5: Form a Perfect Square

Now, recognize \(v^2 + 4v + 4\) as a perfect square trinomial, which is \((v+2)^2\). The equation becomes \((v+2)^2 - 4 = 6\).

Step 6: Solve for the Perfect Square

Add 4 to both sides to isolate the perfect square: \((v+2)^2 = 10\).

Step 7: Solve for v

Take the square root of both sides. This gives us two possible equations: \(v+2 = \sqrt{10}\) and \(v+2 = -\sqrt{10}\).

Step 8: Final Solution

Solving for \(v\) in both equations, we get \(v = \sqrt{10} - 2\) and \(v = -\sqrt{10} - 2\). These are the solutions to the quadratic equation.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). These equations are crucial because they can describe a variety of physical phenomena, from the path of a projectile to the economics of profit maximization.

To solve quadratic equations, we may use several methods, such as factoring, using the quadratic formula, or completing the square. Each method has its own advantages and is chosen based on the specific form of the equation.

Understanding the structure of a quadratic equation helps in deciding the best approach to find its roots, which are the values of \(x\) that satisfy the equation. These roots can be real or complex, and their nature is determined by the discriminant \(b^2 - 4ac\), a key concept when working with quadratic equations.

Perfect Square Trinomial

Recognizing a perfect square trinomial is a key skill in algebra, especially when solving quadratic equations by completing the square.

A perfect square trinomial is an expression like \(x^2 + 2xy + y^2\), which can be factored into \((x + y)^2\). This pattern is based on squaring a binomial. Notice that it follows a specific pattern:

• The first term is a square: \(x^2\).
• The last term is a square: \(y^2\).
• The middle term is twice the product of the terms from the first and the last: \(2xy\).

Identifying perfect square trinomials simplifies solving quadratic equations, as seen with the expression \(v^2 + 4v + 4\), which can be rewritten as \((v+2)^2\). This transformation is central to the technique of completing the square, making the equation easier to handle and solve.

Solving Equations

Solving equations involves finding the values that make an equation true. For quadratic equations like \(v^2 + 4v = 6\), solving entails finding the values of \(v\) that satisfy the equation.

The method of completing the square involves a specific set of steps designed to transform the equation into a form that reveals its solutions:

• Move Constant Terms: Place the constant term on one side to simplify the completing process, as seen when the equation becomes \(v^2 + 4v - 6 = 0\).
• Form a Perfect Square: Add and subtract the same number to create a perfect square trinomial on one side of the equation.
• Isolate the Square: Create an expression that can be written as \((v+2)^2 = 10\), allowing you to solve for \(v\).
• Solve by Extraction: Take the square root of both sides to find the possible values, leading to solutions like \(v = \sqrt{10} - 2\) and \(v = -\sqrt{10} - 2\).

This systematic approach ensures that you can solve any quadratic equation effectively using the completing the square method.