x^2 + \left(-\frac{3}{4}x\right)^2 = 25

["SEO-Optimized Article: Solving the Equation x² + (\left(-\frac{3}{4}x\right)^2 = 25)", "---", "Understanding and Solving the Quadratic Equation: (x^2 + \left(-\frac{3}{4}x\right)^2 = 25)", "If you’ve recently encountered the equation (x^2 + \left(-\frac{3}{4}x\right)^2 = 25), you’re not alone—this type of expression shows up often in algebra and geometry problems. This article breaks down the equation step-by-step, solves it clearly, and explains its real-world and mathematical significance. Perfect for students, educators, and math enthusiasts looking to master quadratic expressions efficiently.", "---", "### Step 1: Simplify the Equation", "The equation contains a squared term with a coefficient:\n[\nx^2 + \left(-\frac{3}{4}x\right)^2 = 25\n]", "First, simplify (\left(-\frac{3}{4}x\right)^2).\nSince squaring eliminates the negative sign:\n[\n\left(-\frac{3}{4}x\right)^2 = \left(\frac{3}{4}\right)^2 \cdot x^2 = \frac{9}{16}x^2\n]", "Rewrite the original equation with this simplification:\n[\nx^2 + \frac{9}{16}x^2 = 25\n]", "---", "### Step 2: Combine Like Terms", "Factor out (x^2) to combine the terms:\n[\nx^2 \left(1 + \frac{9}{16}\right) = 25\n]", "Convert 1 to a fraction with denominator 16:\n[\nx^2 \left(\frac{16}{16} + \frac{9}{16}\right) = 25 \quad \Rightarrow \quad x^2 \left(\frac{25}{16}\right) = 25\n]", "---", "### Step 3: Solve for (x^2)", "Divide both sides by (\frac{25}{16}) (or multiply by its reciprocal (\frac{16}{25})):\n[\nx^2 = 25 \ imes \frac{16}{25} = 16\n]", "---", "### Step 4: Solve for (x)", "Take the square root of both sides:\n[\nx = \pm\sqrt{16} = \pm4\n]", "---", "### Final Answer", "The solutions to the equation (x^2 + \left(-\frac{3}{4}x\right)^2 = 25) are:\n[\n\boxed{x = 4} \quad \ ext{and} \quad \boxed{x = -4}\n]", "---", "### Why This Equation Matters", "Equations like this often appear in:", "- Geometry: When calculating distances using the Pythagorean theorem with linear expressions.\n- Physics: Modeling motion or energy relationships involving squared variables.\n- Calculus & Algebra: As foundational forms for understanding parabolic functions and quadratic forms.", "Understanding how to simplify and solve such expressions strengthens algebraic intuition and prepares learners for advanced mathematical challenges.", "---", "### Practice Tips", "To master solving such equations, try varying the coefficient—experiment with different fractions or integers in place of (-\frac{3}{4}). Also, graph both sides to visualize where the expressions equal 25.", "---", "Key SEO Keywords:\nx² + (−3/4x)² = 25, solving quadratic equations, algebra 1 exercises, simplified radical equations, algebra practice problems, step-by-step khan academy style quadratic, solving equations with exponents and fractions.", "Meta Description:\nA clear, step-by-step guide to solving (x^2 + \left(-\frac{3}{4}x\right)^2 = 25). Learn algebra techniques, simplify expressions, and master solving quadratic equations today.", "---", "Try solving it now—experience the power of algebra!\nMastering equations like this one unlocks deeper insights into mathematics, science, and engineering.", "---", "By breaking down the expression into simple algebraic steps, anyone can solve quadratic forms with confidence. This foundational skill translates into stronger problem-solving abilities across STEM fields."]