Problem 17 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by completing the square. Exercises \(11-14\) and \(17-20\) may be checked by factoring. $$v(v+2)=15$$

Step-by-Step Solution

Answer

The solutions are \( v = 3 \) and \( v = -5 \).

Step 1: Expand the Equation

Start with the equation given: \[ v(v + 2) = 15 \] Expand the left-hand side: \[ v^2 + 2v = 15 \] This is the expanded form of the quadratic.

Step 2: Move Constant Term to One Side

To complete the square, move the constant term to the right-hand side of the equation: \[ v^2 + 2v = 15 \] Becomes: \[ v^2 + 2v - 15 = 0 \] But for completing the square, it's more useful without the constant: \[ v^2 + 2v = 15 \]

Step 3: Prepare to Complete the Square

To complete the square, we need to work with the form \( v^2 + 2v + c \). We will find a value of \(c\) that completes the square. Take half of the coefficient of \(v\) (which is 2), square it, and add and subtract this value from the equation. Half of 2 is 1, and its square is 1.

Step 4: Add and Subtract the Square

Add and subtract 1 inside the equation: \[ v^2 + 2v + 1 - 1 = 15 \] Simplify this to: \[ (v + 1)^2 - 1 = 15 \]

Step 5: Solve for v

Isolate the square term by adding 1 to both sides: \[ (v + 1)^2 = 16 \] Take the square root of both sides: \[ v + 1 = \pm 4 \] Thus, \(v + 1 = 4\) or \(v + 1 = -4\).

Step 6: Solve for Each Case

For \(v + 1 = 4\): Subtract 1 from both sides: \[ v = 3 \] For \(v + 1 = -4\): Subtract 1 from both sides: \[ v = -5 \]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by turning a quadratic expression into a perfect square trinomial. This technique provides an alternative way to solve quadratics when other methods, like factoring, are not immediately clear. Here's how it works:

• First, make sure that the quadratic equation is in the standard form: \[ ax^2 + bx + c = 0 \] however, for completing the square, it's easier to work without the constant on the same side as the quadratic terms.
• Next, take the coefficient of the linear term, divide it by two, and square the result. This squared value is then used to "complete the square."
• Add this squared value inside the quadratic expression, effectively transforming it into a perfect square trinomial \[ (x + d)^2 \].
• Balance the equation by adding or subtracting the same value on both sides, ensuring the equation's equality is maintained.

By following these steps, the equation becomes easier to solve, leading to solutions via further manipulation like taking the square root.

Factoring Quadratic Equations

Factoring quadratic equations is a common method used for finding the roots, or solutions, of a quadratic expression. This method involves expressing the quadratic as a product of two binomials when possible. Factoring is effective when dealing with simple quadratics or those whose solutions are rational numbers.

• Start by writing the quadratic in the standard form \[ ax^2 + bx + c = 0 \].
• Look for two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\).
• If these two numbers can be found, then write the quadratic expression as a product of two binomials: \( (x + m)(x + n) = 0 \).
• Solve each binomial set to zero to find the roots of the equation, \( x = -m \) and \( x = -n \).

The efficiency and simplicity of factoring make it a first choice method, especially under conditions where the equation easily lends itself to this approach.

Solving Quadratic Equations

The ultimate goal when handling quadratic equations is to find the value(s) of the variable that make the equation true. There are several methods robust enough to tackle these equations: completing the square, factoring, and utilizing the quadratic formula. Here's how they contribute:

• Completing the Square: Transforms the quadratic into a perfect square trinomial, useful when the equation doesn't factor easily.
• Factoring: Efficient for simple quadratics with integer solutions, by transforming them into a product of binomials.
• Quadratic Formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It's a universal solution, always reliable for any quadratic equation.

Choosing the right method generally depends on the specific characteristics of the equation to be solved. Some equations lend themselves to easier manipulation and clearer methods, making solving straightforward and effective.

Mathematics Education

Mathematics education plays a crucial role in building the foundation for logical thinking and problem-solving skills. Understanding quadratic equations is a key building block in this learning process. Here's why:

• Conceptual Understanding: Knowing how to solve quadratic equations helps students see the arithmetic process and its application in real-life scenarios.
• Skill Building: In learning how to factor or complete the square, students enhance their algebraic manipulation skills, which are applicable in advanced mathematical topics.
• Confidence Development: Successfully solving such problems boosts student confidence, encouraging them to tackle more complex math topics.
• Real-World Connection: Quadratic equations model a variety of real-world phenomena, from physics to finance, reinforcing the practical value of these mathematical concepts.

In essence, learning quadratic equations not just equips students with mathematical tools, but also enriches their critical thinking and analytical capabilities.