Expanding the Quadratic Expression: x² − x − 6 = 0 — Solving the Equation Step by Step
Solving quadratic equations is a foundational skill in algebra, essential for students and coding enthusiasts alike. One of the most commonly encountered quadratic expressions is x² − x − 6 = 0. Expanding and solving this equation helps strengthen your understanding of factoring, the quadratic formula, and real-world applications. In this article, we’ll walk through the step-by-step expansion, factoring, and solving of the equation x² − x − 6 = 0, providing clear explanations and practical insight.
What is the Quadratic Equation?
A standard quadratic equation takes the form:
ax² + bx + c = 0,
where a, b, and c are constants, and a ≠ 0.
In our example:
- a = 1
- b = −1
- c = −6
Expanding the quadratic allows us to solve for x using factoring, completing the square, or the quadratic formula.
Step 1: Expanding the Quadratic Expression
Although x² − x − 6 is already expanded, understanding expansion helps when you start with a factored form. Suppose you factor it as:
x² − x − 6 = (x − 3)(x + 2)
Now, expand it using the distributive property:
(x − 3)(x + 2) = x·x + x·2 − 3·x − 3·2 = x² + 2x − 3x − 6 = x² − x − 6
This confirms the factorization is correct.
Step 2: Solving x² − x − 6 = 0 by Factoring
Set each factor equal to zero:
(x − 3)(x + 2) = 0
For the product to be zero, either factor must be zero:
- x − 3 = 0 → x = 3
- x + 2 = 0 → x = −2
solutions: x = 3, x = −2
Step 3: Using the Quadratic Formula (Alternative Method)
For any ax² + bx + c = 0, the quadratic formula gives:
x = [ −b ± √(b² − 4ac) ] / (2a)
Plugging in a = 1, b = −1, c = −6:
x = [ −(−1) ± √( (−1)² − 4·1·(−6) ) ] / (2·1)
x = [ 1 ± √(1 + 24) ] / 2
x = [ 1 ± √25 ] / 2
x = [ 1 ± 5 ] / 2
Thus:
- x = (1 + 5)/2 = 6/2 = 3
- x = (1 − 5)/2 = −4/2 = −2
Same solutions as before — verifying our factoring.
Step 4: Graphing the Quadratic Function
The equation corresponds to the parabola:
y = x² − x − 6
- The x-intercepts (roots) are at x = 3 and x = −2, where the graph crosses the x-axis.
- The y-intercept is at y = −6, found by setting x = 0.
- The parabola opens upward because the coefficient of x² is positive.
Real-World Applications
Quadratic equations model many real-life situations, such as:
- Projectile motion in physics (height over time).
- Revenue optimization in business (ep expenditures and pricing).
- Area calculations in architecture and design.
For instance, finding zero crossings of a quadratic can indicate when a business reaches break-even.
Why Learn to Expand and Solve Quadratics?
Mastering the expansion and solving of quadratics like x² − x − 6 = 0 builds a solid foundation for advanced math topics—calculus, linear algebra, engineering, and computer science algorithms involving optimization.
Summary
- x² − x − 6 = 0 is a basic quadratic equation.
- Factoring gives (x − 3)(x + 2) = 0, so solutions are x = 3 and x = −2.
- The quadratic formula confirms these answers reliably.
- Graphically, the roots are the x-intercepts at x = −2 and x = 3.
- Understanding this equation enhances algebraic fluency and problem-solving skills.
Further Reading & Practice
- Try solving other quadratics like x² + 5x + 6 = 0 or x² − 4x = 0.
- Explore graphing tools like Desmos or GeoGebra to visualize the solutions.
- Challenge yourself with word problems involving quadratic relationships.
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Whether you’re a student preparing for exams or a programmer building math-based algorithms, mastering the expansion and solution of x² − x − 6 = 0 opens doors to powerful mathematical thinking. Start expanding — and expanding your knowledge!
