Problem 17 - Basic Technical Mathematics with Calculus

Question

In Exercises \(11-30,\) 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

We begin by expanding the left-hand side of the equation. The equation is \[ v(v + 2) = 15 \]Expanding this, we have:\[ v^2 + 2v = 15 \]

Step 2: Move Constant to One Side

Next, let's move the constant from the right side to the left side of the equation to set it to 0:\[ v^2 + 2v - 15 = 0 \]

Step 3: Organize for Completing the Square

Reorganize the equation to prepare for completing the square.\[ v^2 + 2v = 15 \]

Step 4: Identify Constant to Complete the Square

To complete the square, take the coefficient of the linear term (in this case, 2), divide it by 2, and square it.\[ \left(\frac{2}{2}\right)^2 = 1 \]Add this square to both sides of the equation.

Step 5: Add and Subtract the Square v^2 + 2v = 15

Add and subtract the square value on the left-hand side while balancing by adding to the right:\[ v^2 + 2v + 1 = 15 + 1 \]This makes:\[ v^2 + 2v + 1 = 16 \]

Step 6: Factor the Perfect Square

Now factor the perfect square trinomial on the left:\[ (v + 1)^2 = 16 \]

Step 7: Solve for v

To solve for \( v \), take the square root of both sides:\[ v + 1 = \pm 4 \]This gives two potential solutions when simplified:\[ v + 1 = 4 \] leads to \( v = 3 \) and\[ v + 1 = -4 \] leads to \( v = -5 \).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by making one side of the equation a perfect square trinomial. This step-by-step transformation helps in easily finding the roots of the equation.

To complete the square, you'll follow these steps:

• First, ensure that the coefficient of the quadratic term is 1. If it's not, divide the entire equation by the coefficient.
• Next, take the linear term's coefficient (the term with the variable to the first power), divide it by 2, and then square the result.
• Add this squared value to both sides of the equation, forming a perfect square trinomial on one side.
• The equation can then be written as a square of a binomial. Finally, solve by using the square root principle.

Through completing the square, we adjust the equation into a format that makes it straightforward to find the solutions for the variable.

Factoring Quadratic Equations

Factoring is a method of solving quadratic equations where we express the quadratic as a product of its linear factors. This approach can be used if the equation can be turned into

• a pair of binomials set to zero.

To factor a quadratic equation:

• Start with arranging the equation in the standard form: \( ax^2 + bx + c = 0 \).
• Identify two numbers that multiply to give the product \( ac \) (where \( a \) is the coefficient of \( x^2 \) and \( c \) is the constant term) and add up to \( b \) (the coefficient of \( x \)).
• Split the middle term using these two numbers, and factor by grouping to form two binomial expressions.
• Set each binomial equal to zero and solve for the variable.

Factoring is especially useful when quick solutions are needed, and the polynomial can be easily simplified.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that can be written as the square of a binomial. These occur often when "completing the square" in quadratic equations.

The standard form of a perfect square trinomial looks like this:

• \((a+b)^2 = a^2 + 2ab + b^2\)

In the process of completing the square, a perfect square trinomial is typically formed by taking half of the linear term's coefficient, squaring it, and rearranging the equation to fit this form.

This simplifies the equation greatly, allowing the quadratic equation to be represented as:

• \((x+constant)^2 = something\)

Once in this form, solving becomes much easier as you separate the variable by isolating it as the square root, leading directly to the solution. Perfect square trinomials, thus, play a critical role in making quadratics easier to solve.