["# Solving the Equation: ((15 - 2x)(10 - 2x) = 110) – A Step-by-Step Guide", "S Verema math learners and problem solvers often encounter quadratic equations that require thoughtful expansion, simplification, and solving. One such common yet essential problem is solving the equation:
\n[
\n(15 - 2x)(10 - 2x) = 110
\n]", "This article breaks down how to solve this equation step-by-step, offering clear explanations, practical strategies, and key insights—making it a valuable SEO resource for students, teachers, and algebra enthusiasts.", "---", "## Why This Equation Matters", "The equation ((15 - 2x)(10 - 2x) = 110) is a product of two linear expressions equaling a constant—this forms a quadratic equation upon expansion. Quadratic equations like this arise in algebra, physics, economics, and many real-world applications. Knowing how to solve them fluently boosts mathematical maturity.", "---", "## Step 1: Expand the Left Side", "To simplify, begin by expanding the product using the distributive property (FOIL method):", "[
\n(15 - 2x)(10 - 2x)
\n]", "Multiply each term:
\n- (15 \ imes 10 = 150)
\n- (15 \ imes (-2x) = -30x)
\n- (-2x \ imes 10 = -20x)
\n- (-2x \ imes (-2x) = 4x^2)", "Add them together:
\n[
\n4x^2 - 50x + 150
\n]", "So the equation becomes:
\n[
\n4x^2 - 50x + 150 = 110
\n]", "---", "## Step 2: Bring All Terms to One Side (Quadratic Form)", "Subtract 110 from both sides:
\n[
\n4x^2 - 50x + 150 - 110 = 0
\n]
\n[
\n4x^2 - 50x + 40 = 0
\n]", "---", "## Step 3: Simplify the Quadratic Equation", "Divide every term by 2 to simplify:
\n[
\n2x^2 - 25x + 20 = 0
\n]", "Now you have a standard quadratic equation:
\n[
\n2x^2 - 25x + 20 = 0
\n]", "---", "## Step 4: Solve Using the Quadratic Formula", "Since factoring may be tricky (check discriminant first), use the quadratic formula:
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "For (2x^2 - 25x + 20 = 0):
\n- (a = 2)
\n- (b = -25)
\n- (c = 20)", "### Calculate the Discriminant:
\n[
\n\Delta = b^2 - 4ac = (-25)^2 - 4(2)(20) = 625 - 160 = 465
\n]", "### Compute Solutions:
\n[
\nx = \frac{-(-25) \pm \sqrt{465}}{2 \cdot 2} = \frac{25 \pm \sqrt{465}}{4}
\n]", "This gives two real, irrational solutions:
\n[
\nx = \frac{25 + \sqrt{465}}{4} \quad \ ext{and} \quad x = \frac{25 - \sqrt{465}}{4}
\n]", "---", "## Step 5: Verify the Solutions", "Always test solutions in the original equation ((15 - 2x)(10 - 2x) = 110) to ensure no extraneous values appear—they can arise during expansion.", "---", "## Tips for Quick Handling", "- Use pattern matching: Recognize when the product of two linear terms equals a constant — often leads to quadratics.
\n- Complete the square or use the quadratic formula—both reliable for solving (ax^2 + bx + c = 0).
\n- Simplify fractions and radicals before final presentation.
\n- Check solutions early to avoid propagation of errors.", "---", "## Real-World Context", "This type of equation models situations involving area, profit margins, or conversion scenarios where two diminishing quantities are multiplied. Understanding it deepens analytical thinking in applied mathematics.", "---", "## Key Takeaways", "- Expansion transforms products into usable quadratic forms.
\n- Simplifying coefficients improves manageability.
\n- The quadratic formula solves most quadratics without factoring.
\n- Verification confirms accuracy and builds confidence.", "---", "## FAQs", "Q: Can I solve this without the quadratic formula?
\nA: Yes, you can factor, but it’s often messy—quadratic formula ensures success every time.", "Q: Why simplify coefficients?
\nA: Simplification reduces errors and eases computation.", "Q: Is this equation always solvable?
\nA: Yes—since the discriminant (165 > 0), two real solutions exist.", "---", "## Conclusion", "Mastering equations like ((15 - 2x)(10 - 2x) = 110) equips learners with foundational algebra skills. With clear expansion, simplification, and solutions via the quadratic formula, solving such problems becomes fast and accurate. Whether preparing for school exams or tackling real-world math, this method is a reliable tool.", "---", "### Key Keywords for SEO:
\nsolve quadratic equation \((15 - 2x)(10 - 2x) = 110\), step-by-step solving product of binomials, quadratic formula application, algebraic equations with real solutions, how to expand and solve quadratic equations, algebra homework help product equations.", "---", "By following this guide, anyone can confidently solve ((15 - 2x)(10 - 2x) = 110) and apply these techniques across various math topics. Keep practicing—every equation enhances your problem-solving power!"]
