If the function \( f(x) = 2x^2 - 4x - 6 \) has a root at \( x = 3 \), find the other root.
["Title: Finding the Second Root of the Quadratic Function ( f(x) = 2x^2 - 4x - 6 ) given One Root is ( x = 3 )", "---", "Introduction\nWhen solving quadratic equations, finding both roots is essential for understanding their behavior and applications. If you're given one root of a quadratic function—especially ( x = 3 ) for ( f(x) = 2x^2 - 4x - 6 )—it’s often straightforward to determine the second root using fundamental algebraic properties. This article walks you through the steps to find the missing root of this quadratic function.", "---", "Given:\nThe function:\n[\nf(x) = 2x^2 - 4x - 6\n]\nKnown root:\n[\nx = 3\n]", "---", "Why one root helps find the other\nFor any quadratic function in the form ( ax^2 + bx + c = 0 ), the sum of the roots is given by the formula:\n[\nx_1 + x_2 = -\frac{b}{a}\n]\nThis comes from factoring or using the quadratic root sum identity. Knowing one root allows you to easily find the second using this sum.", "---", "Step 1: Identify coefficients\nFrom ( f(x) = 2x^2 - 4x - 6 ):\n- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "---", "Step 2: Use the sum of roots formula\nSince ( x_1 = 3 ) and ( x_1 + x_2 = -\frac{b}{a} ):\n[\n3 + x_2 = -\frac{-4}{2} = \frac{4}{2} = 2\n]\n[\n3 + x_2 = 2\n]\n[\nx_2 = 2 - 3 = -1\n]", "---", "Alternative: Use the factored form\nSince ( x = 3 ) is a root, ( (x - 3) ) is a factor of ( f(x) ). Let’s factor the quadratic:\n[\nf(x) = 2x^2 - 4x - 6 = 2(x - 3)(x + 1)\n]\nExpanding verifies:\n[\n2(x^2 - 2x - 3) = 2x^2 - 4x - 6\n]\nThe roots are ( x = 3 ) and ( x = -1 ).", "---", "Conclusion\nGiven ( f(x) = 2x^2 - 4x - 6 ) has a root at ( x = 3 ), the other root is ( x = -1 ). This method leverages the elegant relationship between coefficients and roots to quickly identify missing solutions.", "---", "Keywords:\nquadratic equation roots, find second root, solve quadratic function, ( f(x) = 2x^2 - 4x - 6 ), Vieta’s formula, algebraic root finding, quadratic factoring", "---", "Meta Description:\nLearn how to find the second root of ( f(x) = 2x^2 - 4x - 6 ) when one root is known as ( x = 3 ). Explore both sum-of-roots and factoring methods for quick solutions.", "---", "Remember: Once one root is confirmed, use sum of roots or factoring to easily find the other — saving time and ensuring accuracy in quadratic analysis.", "---", "Visit us for more tutorials on solving quadratics, factoring polynomials, and applying Vieta’s formulas!"]
