Problem 15 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by completing the square. $$D^{2}+3 D+2=0$$

Step-by-Step Solution

Answer

The solutions are \( D = -1 \) and \( D = -2 \).

Step 1: Move Constant Term to Right Side

The equation is given as \( D^2 + 3D + 2 = 0 \). First, we need to move the constant term to the right side to prepare for completing the square. We do this by subtracting 2 from both sides, giving us: \[ D^2 + 3D = -2 \]

Step 2: Find the Term for Completing the Square

To complete the square, we need to find the term that will make \( D^2 + 3D \) a perfect square trinomial. Take half of the coefficient of \( D \), which is 3, then square it. \[ \left( \frac{3}{2} \right)^2 = \frac{9}{4} \]

Step 3: Add and Subtract \( \frac{9}{4} \)

Add and subtract \( \frac{9}{4} \) inside the equation to balance it. Add \( \frac{9}{4} \) to both sides:\[ D^2 + 3D + \frac{9}{4} = -2 + \frac{9}{4} \]

Step 4: Simplify Both Sides

The left side is a perfect square trinomial:\( (D + \frac{3}{2})^2 \)Now, simplify the right side:\[ -2 + \frac{9}{4} = -\frac{8}{4} + \frac{9}{4} = \frac{1}{4} \]

Step 5: Solve the Perfect Square Equation

We have:\[ (D + \frac{3}{2})^2 = \frac{1}{4} \] Take the square root of both sides:\[ D + \frac{3}{2} = \pm \frac{1}{2} \]

Step 6: Isolate D and Find Solutions

Solve for \( D \) by isolating it:1. \( D + \frac{3}{2} = \frac{1}{2} \) \( D = \frac{1}{2} - \frac{3}{2} = -1 \)2. \( D + \frac{3}{2} = -\frac{1}{2} \) \( D = -\frac{1}{2} - \frac{3}{2} = -2 \)Thus, the solutions are \( D = -1 \) and \( D = -2 \).

Key Concepts

Completing the Square

"Completing the square" is a technique for solving quadratic equations. It involves manipulating an equation into a form where one side is a perfect square trinomial. This method can help with graphing quadratic functions or finding their roots. Let's go through the process.

• Move the constant to the other side: Start by moving the constant term (the number without a variable) to the other side of the equation. This leaves the variable terms on one side, making it easier to work with.
• Find the magic number: Next, take half of the coefficient of the linear term (the term with the variable, in this case, "D"), and square it. This number will help you transform the existing quadratic expression into a perfect square trinomial.
• Add and subtract the same number: Add the number found in the previous step to both sides of the equation. This balance keeps the equation true while allowing you to reformulate one side into a perfect square trinomial.

Completing the square is handy because it reshapes the quadratic into a format that is much simpler to solve, particularly when the quadratic doesn't factor neatly. Once you know how, it'll feel like a nifty mathematical trick!

Perfect Square Trinomial

Creating a perfect square trinomial is an essential step in the process of completing the square. When you complete the square, your goal is to form an expression that can be rewritten as the square of a binomial.

• The structure: A perfect square trinomial looks like this: \( a^2 + 2ab + b^2 = (a + b)^2 \)
• Identify the squared terms: For \( D^2 + 3D + \frac{9}{4} \), you know \( D^2 \) is the square of \( D \) and \( \frac{9}{4} \) is \( \left(\frac{3}{2}\right)^2 \).
• Check the middle term: Ensure the middle term, \( 3D \), fits the pattern \( 2ab \). Here \( a = D \) and \( b = \frac{3}{2} \). Our expression rewrites as \( (D + \frac{3}{2})^2 \).

Recognizing this structure enables you to simplify significantly, revealing the power of this formulation. Perfect square trinomials are versatile tools within algebra, appearing in many mathematical situations.

Solving Equations

Once you've rewritten a quadratic as a perfect square trinomial, solving for the variable is straightforward. The process effectively reduces the complexity of the equation, allowing for easy solutions.

• Take the square root: With an equation like \( (D + \frac{3}{2})^2 = \frac{1}{4} \), take the square root of both sides. This effectively undoes the squaring, leaving you with a simpler linear equation.
• Isolate the variable: Solve the resulting simple equations by isolating the variable on one side. This typically involves straightforward addition or subtraction.
• Consider both solutions: Don't forget that squaring can conceal sign differences. This means the solution can be both positive and negative values, which could lead to two separate solutions.

By completing the square, the quadratic becomes easier to handle, and you're able to extract solutions that might be less obvious in its original form. This method also makes it easier to understand the geometry and solutions of quadratic equations.