Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ x^{2}+14 x+44=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x1 = -4\) and \(x2 = -10\)

Step 1: Identify a, b, and c

In the given equation \(x^2 + 14x + 44 = 0\), a = 1, b = 14 and c = 44

Step 2: Substitution Into the Quadratic Formula

Substitute a, b, and c into the quadratic formula. You get, \(x = \frac{-14 ± \sqrt{14^2 - 4*1*44}}{2*1}\)

Step 3: Simplify the Calculation

Simplify the calculation under the square root and the denominator: \(x = \frac{-14 ± \sqrt{196 - 176}}{2}\)

Step 4: Calculate the Values of x

Now, calculate the two possible solutions for x. You get: \(x1 = \frac{-14 + \sqrt{20}}{2} = -4\) and \(x2 = \frac{-14 - \sqrt{20}}{2} = -10\)

Key Concepts

Solving Quadratic Equations

In algebra, quadratic equations are a staple and mastering them is crucial for progressing in mathematics. A quadratic equation is any equation that can be rearranged in standard form as \(ax^2+bx+c=0\). Solving these equations can seem daunting at first, but by using techniques like factoring, completing the square, or the Quadratic Formula, it becomes manageable.

• The formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] allows finding the roots of any quadratic equation.
• The quadratic formula works when you can identify the coefficients \(a\), \(b\), and \(c\) in the equation \(ax^2+bx+c=0\).
• It's a reliable tool for solving all types of quadratic equations, especially when they can't be easily factored.

Start by identifying \(a\), \(b\), and \(c\) from your equation and substitute them into the formula. Then, follow the order of operations to simplify and solve for \(x\). Once you master this formula, solving quadratic equations becomes just a series of straightforward steps.

Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. At its core, algebra is all about finding the unknown. It's a bit like solving puzzles. When you have an algebraic equation, your job is to work out what these unknowns are.

• In the context of quadratic equations, algebra helps determine the values of \(x\) that satisfy the equation.
• To solve for \(x\) using algebra, you may need to manipulate the equation by simplifying expressions and isolating terms.
• Algebra provides the language and tools - like the quadratic formula - to express and solve equations systematically.

When working with algebra, pay close attention to each small step, as even simple mistakes can lead you off the path of finding the correct solution. Practice problems regularly to enhance your skills and understanding.

Discriminant in Quadratic Equations

The discriminant is a key concept in solving quadratic equations. It is part of the quadratic formula, represented as \(b^2-4ac\), and plays a crucial role in determining the nature of the roots of the equation. Here's how the discriminant affects the solution:

• If \(b^2-4ac > 0\), the quadratic equation has two distinct real roots. This means there are two different solutions for \(x\).
• If \(b^2-4ac = 0\), the equation has exactly one real root, which is sometimes called a repeated or double root.
• If \(b^2-4ac < 0\), the equation has no real roots but two complex roots instead. This situation occurs when the solution involves the square root of a negative number.

By evaluating the discriminant before solving a quadratic equation, you gain insight into what kind of solutions to expect. Knowing if the roots are real or complex can be incredibly useful when planning your approach to solving the equation.