x^2 + 2x - 3x - 6 = 0 \\ - Deep Underground Poetry
Solving the Quadratic Equation: x² + 2x – 3x – 6 = 0
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.
Understanding the Context
What is the Given Equation?
The equation is:
x² + 2x – 3x – 6 = 0
At first glance, combining like terms simplifies the equation significantly.
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Key Insights
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.
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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.
