\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -50\), \(c = 40\).
["# Solving Quadratic Equations: The Foundation of (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}) with (a = 4), (b = -50), (c = 40)", "Quadratic equations form the backbone of algebra and are fundamental in mathematics, physics, engineering, and many scientific fields. One of the most powerful tools for solving quadratic equations of the form (ax^2 + bx + c = 0) is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula efficiently provides the roots (solutions) of any quadratic equation, whether the solutions are real or complex. In this article, we will explore how this formula works and walk through its application using specific values: (a = 4), (b = -50), and (c = 40).", "## Why the Quadratic Formula Matters", "Quadratic equations frequently appear in modeling real-world situations such as projectile motion, optimization problems, and electrical circuits. The quadratic formula allows us to find exact solutions—when they exist—without relying on trial and error or graphical approximations.", "### The Standard Form and Discriminant", "Before solving, it’s essential to recognize the equation’s structure and the discriminant, given by (D = b^2 - 4ac). This value determines the nature of the roots:", "- If (D > 0): Two distinct real roots.\n- If (D = 0): One real root (a repeated root).\n- If (D < 0): Two complex conjugate roots.", "## Applying the Formula: Step-by-Step Example", "Given:\n(a = 4), (b = -50), (c = 40)", "First, compute the discriminant:", "[\nD = b^2 - 4ac = (-50)^2 - 4(4)(40) = 2500 - 640 = 1860\n]", "Since (D = 1860 > 0), the equation has two distinct real solutions.", "Now apply the quadratic formula:", "[\nx = \frac{-(-50) \pm \sqrt{1860}}{2 \cdot 4} = \frac{50 \pm \sqrt{1860}}{8}\n]", "We simplify (\sqrt{1860}) by factoring:", "[\n1860 = 4 \cdot 465 = 4 \cdot 15 \cdot 31 = 4 \cdot 3 \cdot 5 \cdot 31\n]", "So:", "[\n\sqrt{1860} = \sqrt{4 \cdot 465} = 2\sqrt{465}\n]", "Thus, the solutions become:", "[\nx = \frac{50 \pm 2\sqrt{465}}{8} = \frac{2(25 \pm \sqrt{465})}{8} = \frac{25 \pm \sqrt{465}}{4}\n]", "### Final Solutions", "The two real roots of the quadratic equation are:", "[\nx = \frac{25 + \sqrt{465}}{4} \quad \ ext{and} \quad x = \frac{25 - \sqrt{465}}{4}\n]", "## Numerical Approximation for Better Understanding", "For practical use, we can approximate the decimal values:", "- (\sqrt{465} \approx 21.56)", "So:", "- (x_1 = \frac{25 + 21.56}{4} = \frac{46.56}{4} \approx 11.64)\n- (x_2 = \frac{25 - 21.56}{4} = \frac{3.44}{4} \approx 0.86)", "These approximate values confirm the distinct real roots found algebraically.", "## Interpretation and Real-World Applications", "The solutions represent key points in parabolic motion or optimization models—such as maximum height in projectile motion or break-even points in economics. Using the quadratic formula ensures precise calculation, avoiding pitfalls of estimation.", "## Summary", "- The quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}) reliably solves equations of any real quadratic form.\n- With (a = 4), (b = -50), (c = 40), we find two distinct real roots derived from the discriminant (D = 1860).\n- Precise symbolic expressions offer exact solutions, while numerical approximations help in practical applications.", "Understanding and mastering this formula empowers you to solve complex quadratic problems across science and engineering.", "---", "Keywords: quadratic formula, (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), solving quadratic equations, discriminant, real and complex roots, example with (a=4), (b=-50), (c=40)\nMeta Description: Master the quadratic formula with detailed steps and numerical solutions using (a = 4), (b = -50), (c = 40). Learn how to solve (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}) and apply it in real-world contexts."]
