Solving the Quadratic Equation x² + 12x + 35 = 0: A thorough look
Introduction
Quadratic equations are fundamental in algebra and appear in various fields, from physics to economics. On top of that, the equation x² + 12x + 35 = 0 is a classic example of a quadratic equation in standard form. Understanding how to solve such equations is crucial for students and professionals alike. Plus, this article explores the methods to solve this specific equation, walks through the theoretical background, and provides practical insights into its applications. Whether you're a student struggling with algebra or someone curious about mathematical problem-solving, this guide will equip you with the knowledge to tackle quadratic equations confidently.
Detailed Explanation
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The term "quadratic" comes from the Latin word quadratus, meaning "square," due to the presence of the squared variable. Solving a quadratic equation involves finding the values of x that satisfy the equation, known as the roots or solutions. These roots can be real or complex numbers, depending on the discriminant (b² - 4ac). For the equation x² + 12x + 35 = 0, we have a = 1, b = 12, and c = 35 And it works..
Quadratic equations are important in modeling real-world phenomena. Still, they describe parabolic trajectories, optimize profit functions, and solve geometric problems. Practically speaking, the ability to solve them efficiently is essential for advancing in mathematics and applying it to practical situations. This equation, in particular, serves as an excellent case study for understanding factoring and the quadratic formula, two primary methods for finding solutions.
Step-by-Step Solution
Factoring Method
To solve x² + 12x + 35 = 0 by factoring, we look for two numbers that multiply to 35 (the constant term) and add up to 12 (the coefficient of the linear term). These numbers are 5 and 7, since 5 × 7 = 35 and 5 + 7 = 12. We can rewrite the equation as:
Not the most exciting part, but easily the most useful.
(x + 5)(x + 7) = 0
Setting each factor equal to zero gives the solutions:
- x + 5 = 0 → x = -5
- x + 7 = 0 → x = -7
Quadratic Formula Method
Alternatively, we can use the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)
Plugging in the values a = 1, b = 12, and c = 35:
x = [-12 ± √(12² - 4 × 1 × 35)] / (2 × 1)
x = [-12 ± √(144 - 140)] / 2
x = [-12 ± √4] / 2
x = [-12 ± 2] / 2
This yields two solutions:
- x = (-12 + 2)/2 = -10/2 = -5
- x = (-12 - 2)/2 = -14/2 = -7
Both methods confirm the roots x = -5 and x = -7.
Real-World Applications
Quadratic equations like x² + 12x + 35 = 0 are not just abstract mathematical concepts; they have tangible applications. In physics, they model the motion of objects under gravity. Take this case: if a ball is thrown upward, its height over time can be described by a quadratic equation. In business, quadratic equations help determine maximum profit by analyzing cost and revenue functions.
Consider a company’s profit function P(x) = -x² + 12x + 35, where x represents the number of units sold. That said, setting P(x) = 0 gives x² - 12x - 35 = 0, which is similar in structure to our equation. Solving this would indicate break-even points, crucial for financial planning. Thus, understanding how to solve such equations is vital for strategic decision-making in various industries That's the whole idea..
Scientific and Theoretical Perspective
The quadratic formula is derived from the method of completing the square, a technique that transforms a quadratic equation into a perfect square trinomial. But for any quadratic equation ax² + bx + c = 0, completing the square involves dividing through by a, rearranging terms, and adding a specific value to both sides to form a square. This process leads to the quadratic formula, which is a universal solution method.
The discriminant (b² - 4ac) plays a critical role in determining the nature of the roots. For x² + 12x + 35 = 0, the discriminant is 144 - 140 = 4, which is positive. This indicates two distinct real roots Took long enough..
