["Solving the Quadratic Equation (2w^2 + 5w - 150 = 0): A Complete Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, economists, and anyone working with mathematical modeling. One such important equation is:", "[
\n2w^2 + 5w - 150 = 0
\n]", "In this SEO-optimized article, we’ll walk you through solving this quadratic equation step-by-step, explain key concepts, and discuss how understanding such problems can boost your analytical and problem-solving skills.", "---", "### What Is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[
\nax^2 + bx + c = 0
\n]", "where (a), (b), and (c) are constants, and (a <br/>\ne 0). The equation (2w^2 + 5w - 150 = 0) fits this form with:", "- (a = 2)
\n- (b = 5)
\n- (c = -150)", "The solutions to this equation give the values of (w) that satisfy the relationship, usually called the roots.", "---", "### Why Solve Quadratic Equations?", "Quadratic equations appear in a wide range of applications:", "- Physics: modeling projectile motion
\n- Economics: profit and cost analysis
\n- Engineering: structural design and optimization
\n- Computer graphics: curve fitting and animations", "Understanding how to solve equations like (2w^2 + 5w - 150 = 0) empowers you to find precise solutions for real-world scenarios.", "---", "### Step-by-Step Solution: Using the Quadratic Formula", "The most reliable method for solving (ax^2 + bx + c = 0) is the quadratic formula:", "[
\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Let’s plug in (a = 2), (b = 5), (c = -150):", "1. Compute the discriminant (D)", "[
\n D = b^2 - 4ac = 5^2 - 4(2)(-150) = 25 + 1200 = 1225
\n ]", "A positive discriminant means two distinct real roots.", "2. Find the square root of the discriminant", "[
\n \sqrt{D} = \sqrt{1225} = 35
\n ]", "3. Apply the quadratic formula", "[
\n w = \frac{-5 \pm 35}{2 \ imes 2} = \frac{-5 \pm 35}{4}
\n ]", "4. Calculate both roots", "- First root:", "[
\n w_1 = \frac{-5 + 35}{4} = \frac{30}{4} = 7.5
\n ]", "- Second root:", "[
\n w_2 = \frac{-5 - 35}{4} = \frac{-40}{4} = -10
\n ]", "---", "### Therefore, the solutions are:", "[
\n\boxed{w = 7.5 \quad \ ext{and} \quad w = -10}
\n]", "---", "### Verifying the Solutions", "Plug the values back into the original equation to confirm:", "- For (w = 7.5):", "[
\n 2(7.5)^2 + 5(7.5) - 150 = 2(56.25) + 37.5 - 150 = 112.5 + 37.5 - 150 = 0
\n ]", "- For (w = -10):", "[
\n 2(-10)^2 + 5(-10) - 150 = 2(100) - 50 - 150 = 200 - 50 - 150 = 0
\n ]", "Both values satisfy the equation, confirming their correctness.", "---", "### How to Choose the Right Root?", "Depending on the context — such as a physical length, time, or quantity — you may select only the positive or practically valid root. In this problem, both (7.5) and (-10) are mathematical solutions, but only (7.5) might make sense if modeling a real-world positive quantity.", "---", "### Mastering Quadratic Solutions with This Example", "Solving (2w^2 + 5w - 150 = 0) reinforces key concepts:", "- Understanding the quadratic formula and discriminant
\n- Recall that the sign and value of (a), (b), and (c) affect root nature
\n- Interpretation of both real roots in context
\n- Verification is essential to prevent errors", "Whether you’re a student learning algebra or a professional applying equations in modeling, mastering this process strengthens your foundation in mathematical problem-solving.", "---", "### Related Searches", "- How to solve quadratic equations by factoring
\n- Differences between quadratic formula, completing the square, and factoring
\n- Real-world applications of quadratic equations
\n- Graphing solutions of (2w^2 + 5w - 150 = 0)
\n- Step-by-step guide to quadratic roots", "---", "### Final Thoughts", "Solving (2w^2 + 5w - 150 = 0) yields practical solutions: (w = 7.5) and (w = -10). Understanding how to derive, verify, and apply these roots equips you with vital analytical tools. Keep practicing — algebra is powerful, and mastering quadratics opens the door to advanced mathematics and real-world problem-solving!", "---", "Keywords for SEO:
\nquadratic equation solution, solve (2w^2 + 5w - 150 = 0), quadratic formula, step-by-step quadratic, real roots of quadratic, algebra practice, solve quadratic equation, discriminant meaning, apply quadratic formula, quadratic root calculation.", "---", "Meta Description:
\nLearn how to solve (2w^2 + 5w - 150 = 0) using the quadratic formula. Step-by-step explanation with verified solutions and real-world applications in algebra.", "---", "Optimized for search engines and practical understanding, this guide helps both beginners and experienced learners confidently tackle quadratic equations."]
