Calculate the discriminant: \( 25^2 - 4 \times 1 \times (-84) = 625 + 336 = 961 \).
["# How to Calculate the Discriminant: A Step-by-Step Guide (Including (25^2 - 4 \ imes 1 \ imes (-84) = 961))", "Understanding the discriminant is essential in algebra, especially when analyzing quadratic equations. The discriminant helps determine the nature of the roots—whether they are real and distinct, real and repeated, or complex. In this article, we’ll focus on how to calculate the discriminant using a classic example:\n[\n25^2 - 4 \ imes 1 \ imes (-84) = 625 + 336 = 961\n]\nThis calculation reveals key insights into the quadratic equation (x^2 + 25x - 84 = 0), and we’ll break down every step to make it easy to understand and apply.", "---", "## What is the Discriminant?", "The discriminant (D) of a quadratic equation in standard form (ax^2 + bx + c = 0) is defined as:\n[\nD = b^2 - 4ac\n]\nDepending on the value of (D):\n- If (D > 0): two distinct real roots\n- If (D = 0): one repeated real root\n- If (D < 0): two complex conjugate roots", "---", "## Step-by-Step Calculation: (25^2 - 4 \ imes 1 \ imes (-84))", "### Step 1: Identify coefficients\nGiven the quadratic (x^2 + 25x - 84 = 0),\n- (a = 1) (coefficient of (x^2))\n- (b = 25) (coefficient of (x))\n- (c = -84) (constant term)", "### Step 2: Plug into the discriminant formula\n[\nD = b^2 - 4ac = 25^2 - 4 \ imes 1 \ imes (-84)\n]", "### Step 3: Compute (25^2)\n[\n25^2 = 625\n]", "### Step 4: Compute the product (4ac)\n[\n4 \ imes 1 \ imes (-84) = 4 \ imes (-84) = -336\n]\nBecause it's (-4ac), and (c = -84), this becomes:\n[\n-4 \ imes a \ imes c = -4 \ imes 1 \ imes (-84) = 336\n]", "### Step 5: Combine values into the formula\n[\nD = 625 + 336\n]", "### Step 6: Add the terms\n[\n625 + 336 = 961\n]", "Final Result:\n[\nD = 961\n]", "---", "## What Does This Mean for the Quadratic Equation?", "Since (D = 961 > 0) and a perfect square ((31^2 = 961)), this quadratic equation has two distinct real roots. Using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{D}}{2a} = \frac{-25 \pm \sqrt{961}}{2} = \frac{-25 \pm 31}{2}\n]\nThis yields:\n- (x = \frac{6}{2} = 3)\n- (x = \frac{-56}{2} = -28)", "So the roots are (x = 3) and (x = -28), confirming two real solutions.", "---", "## Why Calculating the Discriminant Matters", "- Predict root nature without solving the equation.\n- Determine if factors are rational or irrational.\n- Guide solution strategies in algebra, calculus, and applications.", "---", "## Conclusion", "Calculating the discriminant is a quick and powerful tool in algebra. In this example, evaluating (25^2 - 4 \ imes 1 \ imes (-84)) yielded (961), confirming two distinct real roots for the quadratic equation. Mastering this step enhances your ability to analyze and solve quadratic expressions efficiently—whether in exams, research, or real-world problems.", "For more tips on solving quadratics and mastering discriminants, keep exploring!", "---", "Keywords: discriminant calculation, quadratic formula, perfect square trinomial, (25^2 - 4ac), solve quadratics, algebra practice, real roots vs complex roots.\nMeta Description: Learn how to calculate the discriminant with a concrete example: (25^2 - 4 \ imes 1 \ imes (-84) = 961). Understand its meaning and impact on quadratic equations."]
