Divide by 2: \( 2n^2 + 5n - 150 = 0 \). Use the quadratic formula:
Solving the Quadratic Equation \(2n^2 + 5n - 150 = 0\) Using the Quadratic Formula
When faced with a quadratic equation like \(2n^2 + 5n - 150 = 0\), using the quadratic formula provides a powerful and reliable method to find exact solutions. Whether you're working on math problems, programming algorithms, or scientific modeling, understanding how to apply this formula is essential. In this article, we’ll break down the step-by-step solution to \(2n^2 + 5n - 150 = 0\) using the quadratic formula and explore its application in real-world scenarios.
What is the Quadratic Formula?
The quadratic formula solves equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are real numbers and \(a \e 0\). The formula is:
\[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula, you can find the two roots (real or complex) of any quadratic equation efficiently.
Step-by-Step Solution to \(2n^2 + 5n - 150 = 0\)
Step 1: Identify coefficientsFrom the equation \(2n^2 + 5n - 150 = 0\), the coefficients are:- \(a = 2\)- \(b = 5\)- \(c = -150\)
Step 2: Calculate the discriminantThe discriminant, \(D\), tells us about the nature of the roots:\[D = b^2 - 4ac\]Substitute the values:\[D = (5)^2 - 4(2)(-150) = 25 + 1200 = 1225\]Since \(D > 0\) and \(D = 1225 = 35^2\), the equation has two distinct real roots.
Step 3: Apply the quadratic formulaNow substitute into the formula:\[n = \frac{-5 \pm \sqrt{1225}}{2 \ imes 2} = \frac{-5 \pm 35}{4}\]
Step 4: Solve for both roots
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First root (\(+\) sign):\[n_1 = \frac{-5 + 35}{4} = \frac{30}{4} = 7.5\]
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Second root (\(-\) sign):\[n_2 = \frac{-5 - 35}{4} = \frac{-40}{4} = -10\]
Final Answer
The solutions to the equation \(2n^2 + 5n - 150 = 0\) are:\[\boxed{n = 7.5 \quad} \ ext{and} \quad n = -10\]
Real-World Applications
Solving quadratic equations like this arises frequently in:- Physics: Calculating motion trajectories under uniform acceleration.- Engineering: Optimizing structural designs with parabolic stress points.- Economics: Modeling revenue functions or cost curves with nonlinear relationships.
Using the quadratic formula ensures accuracy without approximations, critical when precision matters.
Summary
Breaking down \(2n^2 + 5n - 150 = 0\) via the quadratic formula offers a clear, step-by-step approach to finding exact roots. With \(a = 2\), \(b = 5\), and \(c = -150\), the discriminant confirmed two real solutions. Whether for homework, coding, or practical problem-solving, mastering this method equips learners to tackle quadratic challenges confidently.
Keywords: quadratic equation, quadratic formula, solve \(2n^2 + 5n - 150 = 0\), discriminant, algebra, real solutions, math tutorMeta Description: Learn how to solve \(2n^2 + 5n - 150 = 0\) using the quadratic formula with step-by-step calculation and real-world application insights.
