First, find the roots of the equation \( 2x^2 - 9x + 7 = 0 \) using the quadratic formula:

["# Finding the Roots of the Equation ( 2x^2 - 9x + 7 = 0 ) Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, and when direct factoring isn’t obvious, the quadratic formula becomes your most reliable tool. In this article, we’ll explore how to find the roots of the equation\n[\n2x^2 - 9x + 7 = 0\n]\nusing this powerful method. By the end, you’ll understand not just the solution, but also how and why the quadratic formula works—empowering you to tackle similar equations with confidence.", "---", "## What Are Quadratic Equations?", "A quadratic equation is any equation of the form:\n[\nax^2 + bx + c = 0\n]\nwhere ( a ), ( b ), and ( c ) are constants and ( a <br/>\neq 0 ). These equations graph as parabolas, and their two solutions—called roots or solutions—represent where the parabola intersects the x-axis.", "---", "## When Not to Factor: The Need for the Quadratic Formula", "Many quadratic equations factor neatly using integers, but not all do. When ( a <br/>\ne 1 ), factoring can be tricky or impossible with whole numbers. This is where the quadratic formula shines—a universal toolkit applicable to any quadratic equation.", "---", "## The Quadratic Formula: A Universal Solver", "The roots of\n[\nax^2 + bx + c = 0\n]\nare given by:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Let’s break down each part:", "- ( b^2 - 4ac ) is the discriminant; it determines the nature of the roots (real, repeated, or complex).\n- The ( \pm ) symbol means there are typically two distinct solutions: one using ( + ), the other ( - ).\n- The entire expression is divided by ( 2a ) to solve for ( x ).", "---", "## Applying the Formula to ( 2x^2 - 9x + 7 = 0 )", "We now identify the coefficients:\n- ( a = 2 )\n- ( b = -9 )\n- ( c = 7 )", "Plug these into the quadratic formula:", "[\nx = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(2)(7)}}{2(2)}\n]", "Step-by-step simplification:", "1. Simplify the numerator term ( -(-9) ):\n [\n -(-9) = 9\n ]", "2. Compute the discriminant ( b^2 - 4ac ):\n [\n (-9)^2 = 81\n ]\n [\n 4ac = 4 \cdot 2 \cdot 7 = 56\n ]\n [\n \ ext{Discriminant} = 81 - 56 = 25\n ]", "3. Take the square root of 25:\n [\n \sqrt{25} = 5\n ]", "4. Complete the expression:\n [\n x = \frac{9 \pm 5}{4}\n ]", "Now find the two solutions:\n[\nx_1 = \frac{9 + 5}{4} = \frac{14}{4} = \frac{7}{2} = 3.5\n]\n[\nx_2 = \frac{9 - 5}{4} = \frac{4}{4} = 1\n]", "---", "## The Roots: ( x = \frac{7}{2} ) and ( x = 1 )", "Thus, the solutions to ( 2x^2 - 9x + 7 = 0 ) are:\n[\nx = 1 \quad \ ext{and} \quad x = \frac{7}{2}\n]", "---", "## Verifying the Roots", "It’s always good practice to substitute the roots back into the original equation to confirm correctness.", "For ( x = 1 ):\n[\n2(1)^2 - 9(1) + 7 = 2 - 9 + 7 = 0 \quad \ ext{✓}\n]", "For ( x = \frac{7}{2} ):\n[\n2\left(\frac{7}{2}\right)^2 - 9\left(\frac{7}{2}\right) + 7 = 2\left(\frac{49}{4}\right) - \frac{63}{2} + 7\n= \frac{98}{4} - \frac{126}{4} + \frac{28}{4} = \frac{0}{4} = 0 \quad \ ext{✓}\n]", "---", "## Why Understanding Roots Matters", "Finding roots isn’t just about solving equations—it’s about understanding where a quadratic function crosses the x-axis, determining maximum/minimum points, optimizing resources in economics, analyzing projectile motion in physics, and much more. Mastery of the quadratic formula equips you with a critical tool for STEM applications and real-world problem-solving.", "---", "## Final Thoughts", "The quadratic formula transforms the challenge of solving any quadratic equation into a straightforward computation. Whether you’re a student learning algebra or a professional applying math in science or engineering, knowing how to apply this formula empowers you to solve complex problems confidently and efficiently.", "So next time you face ( ax^2 + bx + c = 0 ) with non-trivial coefficients, remember: the quadratic formula is your fastest, most reliable ally.", "---", "Keywords for SEO: quadratic formula, solve quadratic equation, roots of quadratic, find roots of 2x² – 9x + 7 = 0, quadratic equation solutions, discriminant calculation, algebra tutorial, quadratic formula step-by-step, learn quadratic formula", "Meta Description:\nLearn how to find the roots of ( 2x^2 - 9x + 7 = 0 ) using the quadratic formula. Step-by-step solution with verification, key concepts, and real-world applications for students and learners."]