2x² + 2x – 144 = 0 → x² + x – 72 = 0
Solving the Quadratic Equation: How to Solve 2x² + 2x – 144 = 0 Using Simplification (x² + x – 72 = 0)
Solve quadratic equations efficiently with a simple algebraic transformation. This article explores how converting the equation 2x² + 2x – 144 = 0 into a simpler form—x² + x – 72 = 0—makes solving for x much easier. Whether you’re a student, educator, or math enthusiast, this step-by-step guide will help you understand the logic and mechanics behind quadratic simplification for faster and clearer results.
Introduction to Quadratic Equations
Quadratic equations are polynomial equations of the second degree, typically expressed in the standard form:
ax² + bx + c = 0
Common examples include ax² + bx + c = 0, where a ≠ 0. These equations can yield two real solutions, one real solution, or complex roots, depending on the discriminant b² – 4ac. One key strategy to solving quadratics is completing the transformation to standard form—often by simplifying coefficients.
In this post, we focus on the effective trick of dividing the entire equation by 2 to reduce complexity: turning 2x² + 2x – 144 = 0 into x² + x – 72 = 0.
Step-by-Step Conversion: From 2x² + 2x – 144 = 0 to x² + x – 72 = 0
Original Equation: 2x² + 2x – 144 = 0
Divide all terms by 2 (to simplify coefficients): (2x²)/2 + (2x)/2 – 144/2 = 0
Simplifies to: x² + x – 72 = 0
Now the equation is simpler yet retains its core quadratic structure and key constants. The discriminant remains unchanged and easy to compute.
Why Simplify: The Benefits of Reformulating the Equation
Simplifying quadratic equations offers clear advantages:
- Fewer coefficients to track: Smaller numbers reduce arithmetic errors.
- Faster solution processing: Approaches like factoring, completing the square, or the quadratic formula become quicker.
- Clearer insight: Recognizing patterns—such as factorable pairs—becomes more manageable.
In our case, dividing by 2 reveals that the equation relates directly to integers: x² + x = 72 → (x² + x – 72 = 0)
This form perfectly sets up methods like factoring, quadratic formula, or estimation.
Solving the Simplified Equation: x² + x – 72 = 0
Now solve x² + x – 72 = 0 using a few common methods.
Method 1: Factoring (Find Two Numbers That Multiply to –72, Add to 1)
We seek integers p and q such that: p × q = –72 p + q = 1
Testing factor pairs of 72:
- 9 and –8 → 9 × (–8) = –72; 9 + (–8) = 1 ✔️
Thus: x² + x – 72 = (x + 9)(x – 8) = 0
Setting each factor to zero: x + 9 = 0 → x = –9 x – 8 = 0 → x = 8
Method 2: Quadratic Formula
Using the quadratic formula: x = [–b ± √(b² – 4ac)] / (2a)
For x² + x – 72 = 0, coefficients are: a = 1, b = 1, c = –72
Compute discriminant: Δ = b² – 4ac = 1² – 4(1)(–72) = 1 + 288 = 289
Since Δ = 17², square root is exact: x = [–1 ± √289] / 2 = [–1 ± 17] / 2
Solutions: x = (–1 + 17)/2 = 16/2 = 8 x = (–1 – 17)/2 = –18/2 = –9
Final Answer
The solutions to 2x² + 2x – 144 = 0 are: x = 8 and x = –9
A simpler, normalized form—x² + x – 72 = 0—enabled rapid factoring and accurate application of both factoring and quadratic formula methods.
Conclusion: Mastering Quadratic Equations Through Simplification
Transforming 2x² + 2x – 144 = 0 into x² + x – 72 = 0 by dividing through by 2 exemplifies a powerful algebraic strategy. Simplifying not only reduces computational load but also enhances conceptual clarity and solution accuracy.
Key Takeaways:
- Divide coefficients when beneficial to simplify calculations.
- Simplification supports direct application of factoring and formula methods.
- Mastery of basic transformations accelerates problem-solving.
For students and professionals alike, treating quadratic equations with confidence begins with mastering such straightforward algebraic manipulations.
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