3x 2 5x 2 0

Understanding and Solving the Quadratic Equation: 3x² + 5x + 2 = 0

At first glance, the string of characters 3x² + 5x + 2 = 0 might look like a cryptic puzzle or an intimidating line of code. In reality, it is a classic and fundamental expression in algebra known as a quadratic equation. This specific equation is not just an abstract mathematical exercise; it is a gateway to understanding patterns, predicting outcomes, and solving real-world problems in physics, engineering, economics, and beyond. The goal when encountering such an equation is to find the values of the variable x (called the "roots" or "solutions") that make the entire statement true. For 3x² + 5x + 2 = 0, we will discover there are two distinct real numbers that satisfy this condition. Mastering its solution provides a template for tackling a vast family of similar problems and builds essential analytical skills.

Detailed Explanation: What is a Quadratic Equation?

A quadratic equation is any polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. Its standard form is ax² + bx + c = 0, where a, b, and c are known numbers (coefficients), and a ≠ 0. If a were zero, the equation would degrade into a simpler linear equation. In our example, 3x² + 5x + 2 = 0, we can immediately identify the coefficients: a = 3, b = 5, and c = 2. The "= 0" part is crucial; it signifies we are looking for the x-intercepts of the parabola represented by y = 3x² + 5x + 2—the points where the graph crosses the x-axis.

The importance of quadratic equations cannot be overstated. They model any situation involving acceleration, area calculations, projectile motion, profit maximization, and even the shape of satellite dishes. The solutions to these equations tell us critical information: when will a ball hit the ground? What dimensions yield the maximum area? At what price is profit zero? Therefore, learning to solve them is a core competency in quantitative reasoning. There are several methods: factoring, completing the square, and using the quadratic formula. For our specific equation, factoring is the most efficient and insightful first approach.

Step-by-Step Breakdown: Solving by Factoring

Factoring is the process of breaking down the quadratic expression into a product of two binomials. The logic is based on the zero-product property: if A × B = 0, then either A = 0 or B = 0 (or both). Our strategy is to rewrite 3x² + 5x + 2 as (px + q)(rx + s) = 0. Let's solve 3x² + 5x + 2 = 0 step-by-step.

Step 1: Identify a, b, c and check for a common factor. Here, a=3, b=5, c=2. There is no common factor across all three terms (3, 5, and 2 share no common divisor other than 1), so we proceed to factor the trinomial directly.

Step 2: Find two numbers that multiply to (a × c) and add to b. This is the "AC method" or "splitting the middle term." We need two numbers whose product is ac = 32 = 6 and whose sum is b = 5. After considering the factor pairs of 6 (1 and 6, 2 and 3, -1 and -6, -2 and -3), we see that 2 and 3 satisfy both conditions: 2 × 3 = 6 and 2 + 3 = 5.

Step 3: Rewrite the middle term (5x) using these two numbers. We split 5x into 2x + 3x. The equation now becomes: 3x² + 2x + 3x + 2 = 0

Step 4: Factor by grouping. Group the first two terms and the last two terms: (3x² + 2x) + (3x + 2) = 0 Now, factor out the greatest common factor (GCF) from each group. From the first group, x is common: x(3x + 2). From the second group, 1 is the GCF: 1(3x + 2). Notice that (3x + 2) is now a common factor: x(3x + 2) + 1(3x + 2) = 0 We can factor out (3x + 2): (3x + 2)(x + 1) = 0

Step 5: Apply the zero-product property. Set each factor equal to zero:

1. 3x + 2 = 0 → 3x = -2 → x = -2/3
2. x + 1 = 0 → x = -1

Step 6: Verify the solutions. Substitute x = -2/3 into the original equation: 3(-2/3)² + 5(-2/3) + 2 = 3(4/9) - 10/3 + 2 = 12/9 - 30/9 + 18/9 = (12 - 30 + 18)/9 = 0/9 = 0. ✓ Substitute x = -1: 3(-1)² + 5(-1) + 2 = 3(1) - 5 + 2 = 3 - 5 + 2 = 0. ✓ Both are correct. Therefore, the