x² + 2x = 140

February 22, 2026

Solving the Quadratic Equation: x² + 2x = 140

Solving quadratic equations is a fundamental skill in algebra, widely applicable in science, engineering, economics, and everyday problem-solving. One commonly encountered equation is x² + 2x = 140. In this article, we’ll walk through the step-by-step process of solving this equation, explain its real-world applications, and provide tips to tackle similar problems efficiently.

What Is the Equation x² + 2x = 140?

The equation x² + 2x = 140 is a first-order quadratic equation (after rearranging) used to model a variety of real-life situations—from geometry and physics to business and finance.

Step 1: Rearranging into Standard Form

To solve, we first bring all terms to one side to establish the standard quadratic form: x² + 2x – 140 = 0

This takes us to the general quadratic equation: ax² + bx + c = 0, where:

a = 1
b = 2
c = –140

Step 2: Solving the Equation

There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula. Let’s explore the most efficient one here.

Method 1: Rearranging and Factoring (if possible)

We look to factor the quadratic expression: x² + 2x – 140 = 0

We need two numbers that multiply to –140 and add to 2. After testing factor pairs, we find: (14) × (–10) = –140, and 14 + (–10) = 4 ❌ 34 + (–32) = 2 ✅

Wait — no pair exactly adds to 2. Try completing the square or use the quadratic formula — more reliable for any quadratic.

Method 2: Using the Quadratic Formula

Since factoring is tricky here, apply the quadratic formula: x = [–b ± √(b² – 4ac)] / (2a)

Plug in a = 1, b = 2, c = –140: x = [ –2 ± √(2² – 4(1)(–140)) ] / (2×1) x = [ –2 ± √(4 + 560) ] / 2 x = [ –2 ± √564 ] / 2

Now simplify √564: √564 = √(4 × 141) = 2√141

Thus, x = [ –2 ± 2√141 ] / 2 x = –1 ± √141

Final Solutions

The two real solutions are: x = –1 + √141 and x = –1 – √141

Approximating √141 ≈ 11.87,

x ≈ –1 + 11.87 = 10.87
x ≈ –1 – 11.87 = –12.87

Real-Life Applications of the Equation x² + 2x = 140

This type of quadratic arises in various practical situations:

1. Geometry — Finding Dimensions

Suppose you have a rectangular garden where the length is 2 meters more than the width, and the area is 140 m². Let width = x → length = x+2. Then, x(x + 2) = 140 → x² + 2x – 140 = 0 — exactly our equation!

2. Projectile Motion

In physics, the height of a projectile at time t may follow a quadratic model. Finding maximum height or time of flight often involves solving such equations.

3. Economics — Profit Maximization

Businesses use quadratic models to predict revenue or cost functions to determine break-even points or maximum profit.

Tips for Solving Quadratic Equations Like This

Always rearrange to standard form: ax² + bx + c = 0
Factor when convenient, but don’t force it—use the quadratic formula if needed
Recognize that discriminant (b² – 4ac) determines the nature of roots
Interpret solutions in context—are they positive, real, or irrelevant?
Use graphing technology or calculators to verify your solutions

Conclusion

The equation x² + 2x = 140 may look simple but opens a world of mathematical modeling. By rearranging, applying algebraic techniques, and interpreting results, you gain skills essential for solving real-world problems. Whether designing a space, calculating motion, or analyzing profits, mastering quadratics puts you ahead.