["# Solving the Quadratic Equation \( w(2w + 5) = 150 \): Step-by-Step Guide", "If you're tackling algebraic equations, quadratic equations like \( w(2w + 5) = 150 \) are common challenges that many students and self-learners face. Solving this equation not only helps strengthen your understanding of quadratic relationships but also sharpens algebraic manipulation skills. In this SEO-optimized article, we will break down how to solve \( w(2w + 5) = 150 \), explain its real-world applications, and provide helpful tips for mastering such quadratic problems.", "## Understanding the Equation", "The given equation is:", "\[
\nw(2w + 5) = 150
\n\]", "This is a first-degree times a linear expression set equal to a constant. To solve for \( w \), we must first expand and convert this into standard quadratic form.", "### Step 1: Expand the Left Side", "Multiply \( w \) through the parentheses:", "\[
\nw \cdot 2w + w \cdot 5 = 2w^2 + 5w
\n\]", "So, the equation becomes:", "\[
\n2w^2 + 5w = 150
\n\]", "### Step 2: Move All Terms to One Side", "Subtract 150 from both sides to form a standard quadratic equation:", "\[
\n2w^2 + 5w - 150 = 0
\n\]", "Now we have:", "\[
\n2w^2 + 5w - 150 = 0
\n\]", "This is the standard quadratic equation \( ax^2 + bx + c = 0 \), with \( a = 2 \), \( b = 5 \), and \( c = -150 \).", "---", "## Solving the Quadratic Equation", "### Option 1: Factoring (if possible)", "Try factoring \( 2w^2 + 5w - 150 \). However, this trinomial doesn’t factor nicely into integers, so factoring alone is impractical here.", "### Option 2: Quadratic Formula", "Use the quadratic formula:", "\[
\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n\]", "Plug in \( a = 2 \), \( b = 5 \), \( c = -150 \):", "\[
\nw = \frac{-5 \pm \sqrt{5^2 - 4(2)(-150)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 1200}}{4} = \frac{-5 \pm \sqrt{1225}}{4}
\n\]", "Since \( \sqrt{1225} = 35 \), we get:", "\[
\nw = \frac{-5 \pm 35}{4}
\n\]", "This gives two solutions:", "\[
\nw = \frac{-5 + 35}{4} = \frac{30}{4} = 7.5
\n\]
\n\[
\nw = \frac{-5 - 35}{4} = \frac{-40}{4} = -10
\n\]", "---", "## Verifying the Solutions", "Plug both values back into the original equation \( w(2w + 5) = 150 \):", "- For \( w = 7.5 \):", "\[
\n7.5(2 \cdot 7.5 + 5) = 7.5(15 + 5) = 7.5 \ imes 20 = 150
\n\]", "Valid.", "- For \( w = -10 \):", "\[
\n-10(2 \cdot -10 + 5) = -10(-20 + 5) = -10 \ imes (-15) = 150
\n\]", "Also valid.", "---", "## Final Answer", "The solutions to the equation \( w(2w + 5) = 150 \) are:", "\[
\n\boxed{w = 7.5 \quad} \ ext{or} \quad w = -10
\n\]", "---", "## Real-World Applications", "Equations like this often appear in real-life modeling scenarios:", "- Business planning: Calculating break-even points where revenue depends on quantity in a quadratic form.
\n- Physics: Modeling motion and projectile trajectories under acceleration.
\n- Engineering design: Optimizing areas, volumes, or structural dimensions.", "---", "## Why Mastering Quadratics Matters for SEO", "Searchers looking for algebra help often use keywords like:
\n- “how to solve \( w(2w + 5) = 150 \)”
\n- “quadratic equation solutions”
\n- “algebra quadratic formula examples”
\n- “step-by-step solve \( 2w^2 + 5w - 150 = 0 \)”", "Writing clear, structured content around these terms improves visibility and helps users find your guide faster.", "---", "## Summary Checklist: Solving \( w(2w + 5) = 150 \)", "1. Expand: \( 2w^2 + 5w = 150 \)
\n2. Rearrange: \( 2w^2 + 5w - 150 = 0 \)
\n3. Apply quadratic formula:
\n \[
\n w = \frac{ -5 \pm \sqrt{1225} }{4} = \frac{-5 \pm 35}{4}
\n \]
\n4. Solutions:
\n \[
\n w = 7.5 \quad \ ext{and} \quad w = -10
\n \]
\n5. Verify both solutions satisfy the original equation.", "---", "## Further Learning Resources", "- Watch video tutorials on “Quadratic Equations Using Formula”
\n- Practice similar problems: \( aw(w + b) = c = 200 \), etc.
\n- Use graphing calculators to visualize parabolas and roots", "---", "Mastering quadratics like \( w(2w + 5) = 150 \) equips you with problem-solving skills essential in academics and beyond. With clear steps, substitution, and real-world relevance, this topic stands strong in SEO-driven educational content.", "Have you solved similar equations? Leave a comment below and share your experience!", "---", "Keywords for SEO:

QuadraticEquations #SolveQuadratics #AlgebraHelp #QuadraticFormula #EquationSolutions #2w2 + 5w = 150 #AlgebraFactoring #MathTutorial #StudentResources", "---", "Revise as needed, add internal links to related topics (e.g., quadratic formula explanations), and ensure alt text on visuals includes keywords like “quadratic equation graph” or “step-by-step solve 2w(2w + 5) = 150”."]