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.

Understanding the Context

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

Key Insights

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)

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Final Thoughts

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