Compute the discriminant: \((-8)^2 - 4 imes 2 imes 6 = 64 - 48 = 16\).

["# Compute the Discriminant: A Step-by-Step Guide Using ((-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16)", "Understanding quadratic equations is fundamental in algebra, and one key step in solving them is computing the discriminant. The discriminant helps determine the nature of the roots—whether they are real and distinct, real and equal, or complex. In this article, we’ll explore how to compute the discriminant using the specific example:", "[\nD = (-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16\n]", "This guide will walk you through calculating the discriminant using both precise computation and clear algebraic logic, supported by the proven formula:\n[\nD = b^2 - 4ac\n]", "---", "## What Is the Discriminant?", "For any quadratic equation in the standard form:", "[\nax^2 + bx + c = 0\n]", "the discriminant (D = b^2 - 4ac) determines the type of roots:", "- If (D > 0): Two distinct real roots\n- If (D = 0): Exactly one real root (real and repeated)\n- If (D < 0): Two complex conjugate roots", "---", "## Step-by-Step Computation", "### Step 1: Identify coefficients (a), (b), and (c)", "In our example equation ((-8)^2 - 4 \cdot 2 \cdot 6):", "- The coefficient (a = -8) (since the equation starts as (-8x^2 + 0x + 32) (implied after expansion), but here manually given as ((-8)^2 - 4 \cdot 2 \cdot 6), so interpret (a = -8))\n- (b = 0) (absent middle term term)\n- (c = 6)", "However, note the original expression:\n[\n(-8)^2 - 4 \cdot 2 \cdot 6\n]\nHere, it appears (a = -8), (b = 0), and (c = 6), matching the standard form with (a) as the leading coefficient.", "### Step 2: Plug into the discriminant formula", "[\nD = b^2 - 4ac = (0)^2 - 4(-8)(6)\n]\n[\nD = 0 - 4(-48) = 0 + 192 = 192 \quad \ ext{(Wait! — re-check!)}\n]", "But our original expression says:\n[\n(-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16\n]", "This suggests reinterpreting: likely, the expression refers to a specific quadratic derived from coefficients (a = -8), (b = 2), (c = 6), not a purely expanded form.", "Let’s verify the full quadratic:\nGiven (a = -8), (b = 2), (c = 6), then:", "[\nD = b^2 - 4ac = (2)^2 - 4(-8)(6) = 4 + 192 = 196\n]\nStill not 16.", "Wait — the original expression is:", "[\n(-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16\n]", "This matches (b = -8), (a = 2), (c = 6)? Let’s double-check the structure.", "Actually, the expression:", "[\n(-8)^2 - 4 \cdot 2 \cdot 6\n]", "follows from setting (a = 2), (b = -8), (c = 6) — no, (b^2 = (-8)^2 = 64), and (4ac = 4 \cdot 2 \cdot 6 = 48), so:", "[\nD = 64 - 48 = 16\n]", "But standard form requires the coefficient of (x) is (b), so the equation would be:", "[\n2x^2 - 8x + 6 = 0\n]", "But the discriminant as computed from coefficients:", "[\nD = (-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16\n]", "Yes — here (b = -8), but (b^2 = 64), still valid. So whether (b = -8) or (8), squaring gives 64. Therefore, the discriminant computation is:", "[\n64 - 48 = 16\n]", "This confirms the correct method.", "---", "## The Bifurcation in Roots: Why (D = 16) Matters", "Since (D = 16 > 0), this quadratic has:", "- Two distinct real roots\n- The roots are rational because the discriminant is a perfect square:\n[\nx = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{16}}{2 \cdot 2} = \frac{8 \pm 4}{4}\n]", "Calculating:", "[\nx_1 = \frac{8 + 4}{4} = \frac{12}{4} = 3, \quad x_2 = \frac{8 - 4}{4} = \frac{4}{4} = 1\n]", "Indeed, the roots are (x = 3) and (x = 1), both real and rational.", "---", "## Pro Tips for Quick Discriminant Calculation", "- Always identify coefficients (a), (b), (c) correctly, even when values appear internal or implied.\n- Square the term independently before multiplication: ((-8)^2 = 64), not just (-8^2).\n- Use parentheses carefully: (b^2) means square of (b), not nested evaluation.\n- Remember: negatives cancel in (b^2) but affect products like (4ac) if (a) or (c) is negative.", "---", "## Real-World Applications", "Understanding the discriminant is not just academic:", "- Engineering: Predicting system stability via quadratic models\n- Physics: Analyzing projectile motion and motion under constraints\n- Economics: Finding break-even points in cost-revenue analysis\n- Computer Graphics: Solving for intersection points of curves", "---", "## Summary", "Computing the discriminant by directly evaluating:", "[\nD = b^2 - 4ac\n]", "as demonstrated with:", "[\n(-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16\n]", "allows us to confidently determine the nature of quadratic roots. When (D > 0), real and distinct roots exist—essential knowledge across science, math, and engineering fields.", "Master this technique, and you unlock deeper insight into the very behavior of quadratic relationships.", "---", "Tags: #Discriminant #QuadraticEquations #Algebra #MathGuide #RootType #QuadraticFormula #LearnMath", "Meta Description: Compute the discriminant using ((-8)^2 - 4 \cdot 2 \cdot 6 = 64 - 48 = 16). Understand real distinct roots and apply this key algebra tool in quadratic analysis."]