Solving the Quadratic Equation: x² + 2x – 3x – 6 = 0

If you’re sitting down to solve the equation x² + 2x – 3x – 6 = 0, you’re already tackling a familiar quadratic expression—but one that may seem tricky at first glance. In this SEO-optimized guide, we’ll walk through simplifying, solving, and understanding the roots of this quadratic equation. Whether you're a high school student, a math enthusiast, or just learning algebra, this article will help you master the problem step by step.

What is the Given Equation?

The equation is:
x² + 2x – 3x – 6 = 0

At first glance, combining like terms simplifies the equation significantly.

Step 1: Simplify the Equation

Combine the linear terms:
2x – 3x = –x

So the equation becomes:
x² – x – 6 = 0

This simplified form, x² – x – 6 = 0, is a standard quadratic equation ready for factoring, completing the square, or using the quadratic formula.

Step 2: Solve by Factoring

To factor x² – x – 6, look for two numbers that multiply to –6 and add up to –1.
These numbers are –3 and +2, since:
–3 × 2 = –6
–3 + 2 = –1

Thus, the factored form is:
(x – 3)(x + 2) = 0

Step 3: Apply the Zero Product Property

If a product equals zero, one of the factors must be zero:
x – 3 = 0 → x = 3
x + 2 = 0 → x = –2

Solutions:
x = 3
x = –2

Why This Equation Matters

Understanding how to combine like terms and factor quadratics is essential in algebra. The roots x = 3 and x = –2 represent the x-intercepts of the corresponding parabola, helping visualize quadratic behavior in graphs, physics, engineering, and economics.

Alternative Methods: Quadratic Formula

Need to confirm your answers or if factoring is challenging? Use the quadratic formula:
x = [–b ± √(b² – 4ac)] / (2a)

From x² – x – 6 = 0, we identify:
a = 1, b = –1, c = –6

Plug in values:
x = [1 ± √(1 + 24)] / 2 = [1 ± √25] / 2 = [1 ± 5] / 2

So:
x = (1 + 5)/2 = 6/2 = 3
x = (1 – 5)/2 = –4/2 = –2

Consistent with factoring—proof that mastering both methods strengthens your algebra skills.

Final Thoughts

The equation x² + 2x – 3x – 6 = 0 simplifies elegantly to x² – x – 6 = 0, which factors neatly to yield solutions x = 3 and x = –2. By combining algebraic simplification with standard solving techniques, you gain not just an answer—but deeper insight into quadratic behavior.

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• How to simplify quadratic equations
  
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Summary:

• Simplify: x² – x – 6 = 0
  
• Factor: (x – 3)(x + 2) = 0
  
• Solutions: x = 3 and x = –2
  
• Use quadratic formula for verification

Master this equation, and boost your confidence in algebra!