Problem 37 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ x^{2}+2 x=15 $$

Step-by-Step Solution

Answer

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

Step 1: Move All Terms to One Side

We need to rewrite the equation in the standard form of a quadratic equation, which is \( ax^{2} + bx + c = 0 \). Start by subtracting 15 from both sides to get: \[ x^{2} + 2x - 15 = 0 \]

Step 2: Identify Coefficients

In the equation \(x^{2} + 2x - 15 = 0\), the coefficients are: - \(a = 1\) (coefficient of \(x^2\)) - \(b = 2\) (coefficient of \(x\)) - \(c = -15\) (constant term)

Step 3: Check for Factoring Possibility

To determine if the equation can be factored, we need two numbers whose product is \(-15\) (the constant term) and whose sum is \(2\) (the coefficient of \(x\)). These numbers are \(5\) and \(-3\).

Step 4: Factor the Quadratic Equation

Using the numbers from Step 3, we can factor the equation as follows: \[ (x + 5)(x - 3) = 0 \] The factored form is obtained by using the numbers that we found: \((x + 5)\) and \((x - 3)\).

Step 5: Solve for x

Using the Zero Product Property, set each factor equal to zero and solve for \(x\): 1. \(x + 5 = 0 \rightarrow x = -5\) 2. \(x - 3 = 0 \rightarrow x = 3\) Thus, the solutions are \(x = -5\) and \(x = 3\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a powerful method to solve quadratic equations when they can be expressed as a product of two binomial expressions. In this method, the goal is to find two numbers that multiply to the constant term, and add up to the coefficient of the linear term. For example, in the equation \[x^2 + 2x - 15 = 0,\]we look for two numbers whose product is \(-15\)and whose sum is \(2\).These numbers are \(5\) and \(-3\),so the equation can be factored as\[(x + 5)(x - 3) = 0.\] Factoring allows us to break down the quadratic equation into simpler components, making it easier to find the solutions. It is important to ensure that the quadratic is in standard form \(ax^2 + bx + c = 0\)before attempting to factor. Not all quadratic equations can be factored using this method, especially when the numbers do not lead to integers.

Zero Product Property

The Zero Product Property is a critical aspect of solving quadratic equations once they have been factored. This property states that if a product of two factors equals zero, then at least one of the factors must be zero. Let's use our factored equation:\[(x + 5)(x - 3) = 0.\]Applying the Zero Product Property, we will set each factor to zero and solve for \(x\):

• \(x + 5 = 0\) leads to \(x = -5\),
• \(x - 3 = 0\) leads to \(x = 3\).

Thus, these are our solutions. The Zero Product Property simplifies our task by reducing a quadratic equation into two linear equations. Each gives a straightforward solution that satisfies the original quadratic equation.

Quadratic Formula

The Quadratic Formula is a universal method applicable to solve any quadratic equation, even when factoring is not possible or straightforward. It is especially useful when the quadratic does not easily factor into integers. The formula is derived from the standard form of a quadratic equation, \(ax^2 + bx + c = 0\),and is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula uses the coefficients \(a\), \(b\),and \(c\)from the equation. In the equation we explored, \(x^2 + 2x - 15 = 0\),if we weren't able to factor it, the quadratic formula could have been used to find \(x\).Substitute:

• \(a = 1\),
• \(b = 2\),
• \(c = -15\).

By placing these values into the formula, we confirm the solutions obtained by factoring: \(-5\) and \(3\).The discriminant, \(b^2 - 4ac\),within the formula helps in determining the nature of the roots: whether they are real or complex. Thus, the Quadratic Formula not only helps solve equations but also provides insights into their characteristics.