Solve the quadratic equation using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = 50\), \(c = -136\).
["Solve Quadratic Equations Made Easy: How to Use the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, widely used in science, engineering, and everyday math problems. For those who want a reliable, step-by-step method, the quadratic formula provides a powerful and efficient solution. In this article, we’ll walk through solving the equation (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}) with concrete values: (a = 4), (b = 50), and (c = -136).", "---", "### What Is the Quadratic Formula?", "The quadratic formula solves equations of the form:", "[\nax^2 + bx + c = 0\n]", "Where:\n- (a), (b), and (c) are real numbers,\n- (a <br/>\neq 0),\n- The term under the square root, (b^2 - 4ac), is called the discriminant, determining the nature of the solutions (real and distinct, real and repeated, or complex).", "The general solution is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "---", "### Step-by-Step: Plug in Your Values", "Given:\n- (a = 4)\n- (b = 50)\n- (c = -136)", "First, compute the discriminant (D = b^2 - 4ac):", "[\nD = (50)^2 - 4(4)(-136)\n]\n[\nD = 2500 - (16)(-136)\n]\n[\nD = 2500 + 2176 = 4676\n]", "Since (D = 4676 > 0), there are two distinct real roots.", "---", "### Calculate the Roots Using the Quadratic Formula", "Now, substitute (a), (b), (c), and (D) into the formula:", "[\nx = \frac{-50 \pm \sqrt{4676}}{2 \cdot 4} = \frac{-50 \pm \sqrt{4676}}{8}\n]", "We simplify (\sqrt{4676}). Approximating:", "[\n\sqrt{4676} \approx 68.38\n]", "So the two solutions are:", "[\nx_1 = \frac{-50 + 68.38}{8} = \frac{18.38}{8} \approx 2.2975\n]\n[\nx_2 = \frac{-50 - 68.38}{8} = \frac{-118.38}{8} \approx -14.7975\n]", "---", "### Final Answer", "The solutions to the equation (4x^2 + 50x - 136 = 0) are approximately:", "[\nx_1 \approx 2.30 \quad \ ext{and} \quad x_2 \approx -14.80\n]", "So, the exact form using the quadratic formula is:", "[\nx = \frac{-50 \pm \sqrt{4676}}{8}\n]", "And numerically, the roots are:", "[\n\boxed{x \approx 2.30 \quad \ ext{and} \quad x \approx -14.80}\n]", "---", "### Why Learn the Quadratic Formula?", "It’s a go-to method for solving any quadratic equation reliably and consistently. Whether you're working on physics problems, optimizing functions, or tackling advanced math, mastering this technique empowers you with a core algebraic tool.", "If you'd like to verify your results, use graphing tools or algebra apps to confirm these approximate decimal values match the exact form given by the quadratic formula. Training your brain to apply this formula step-by-step becomes faster with practice—and your math confidence will grow!", "---", "Keywords: quadratic formula, solve quadratic equations, quadratic formula steps, solve (4x^2 + 50x - 136 = 0), discriminant (b^2 - 4ac), real quadratic roots, algebra tutorial, quadratic solutions."]
