The quadratic \( 2x^2 - 9x + 7 \) opens upwards (as \( a > 0 \)), so it is negative between its roots, \( 1 < x < 3.5 \).

["Title: Understanding When the Quadratic ( 2x^2 - 9x + 7 ) is Negative—Roots and Behavior", "---", "Introduction", "Understanding the sign of a quadratic function—whether it is positive, negative, or zero—helps in solving inequalities, modeling real-world situations, and analyzing graph behavior. A key example is the quadratic ( f(x) = 2x^2 - 9x + 7 ), which opens upwards (since the leading coefficient ( a = 2 > 0 )). In this article, we explore why this quadratic is negative specifically between its two real roots: ( 1 < x < 3.5 ).", "---", "1. The Shape of the Quadratic: Upwards-Opening Parabola", "Because the coefficient of ( x^2 ), which is ( a = 2 ), is positive, the parabola opens upward. This means the function takes negative values between its two vertex-intercepts (roots) and positive values outside this interval.", "Graphical Insight:\n- The parabola dips below the x-axis between the roots.\n- Above the x-axis outside the roots.\n- Touches at the roots — zero at them.", "---", "2. Finding the Roots: Where the Function Crosses the x-Axis", "To determine where ( f(x) = 2x^2 - 9x + 7 ) is negative, we first solve for the roots by setting the equation to zero:", "[\n2x^2 - 9x + 7 = 0\n]", "Apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 2 ), ( b = -9 ), ( c = 7 ):", "[\nx = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(2)(7)}}{2(2)} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}\n]", "So the roots are:", "[\nx = \frac{9 + 5}{4} = \frac{14}{4} = 3.5 \quad \ ext{and} \quad x = \frac{9 - 5}{4} = \frac{4}{4} = 1\n]", "Roots: ( x = 1 ) and ( x = 3.5 )", "---", "3. Determining the Sign Between the Roots", "Since the parabola opens upward:", "- The function is zero at the roots (( x = 1 ) and ( x = 3.5 )).\n- It is negative between the roots because the graph lies below the x-axis there.\n- It is positive when ( x < 1 ) or ( x > 3.5 ).", "Thus, the inequality ( 2x^2 - 9x + 7 < 0 ) holds only when:", "[\n1 < x < 3.5\n]", "---", "4. Verifying with Test Values", "Pick test points in the intervals to confirm:", "- For ( x = 2 ) (between 1 and 3.5):\n ( f(2) = 2(2)^2 - 9(2) + 7 = 8 - 18 + 7 = -3 < 0 ) ✅", "- For ( x = 0 ) (less than 1):\n ( f(0) = 0 - 0 + 7 = 7 > 0 ) ✅", "- For ( x = 4 ) (greater than 3.5):\n ( f(4) = 2(16) - 36 + 7 = 32 - 36 + 7 = 3 > 0 ) ✅", "Confirmed: Negative only between ( x = 1 ) and ( x = 3.5 ).", "---", "5. Summary: Key Takeaways", "- The quadratic ( 2x^2 - 9x + 7 ) opens upwards (( a > 0 )).\n- It crosses the x-axis at ( x = 1 ) and ( x = 3.5 ).\n- Between these roots, the graph lies below the x-axis, so ( f(x) < 0 ).\n- Outside this interval, it is positive.\n- This interval ( 1 < x < 3.5 ) is where the quadratic is negative.", "---", "Conclusion", "Understanding that a positive-leading-coefficient quadratic is negative between its conjugate (real) roots provides a powerful tool for solving inequalities. For ( 2x^2 - 9x + 7 ), this interval ( (1, 3.5) ) captures exactly where the function dips below zero. Knowing this helps students confidently analyze and graph quadratics.", "---", "Related Keywords for SEO: \nQuadraticFunctions #ParabolaAnalysis #2x2Quadratic #QuadraticInequalities #RootsOfQuadratic #NegativeBetweenRoots #Upward Opening Parabola #SolvingQuadratics #AlgebraTutorial", "---", "Meta Description:\nDiscover why ( 2x^2 - 9x + 7 ) is negative between ( x = 1 ) and ( x = 3.5 ). Learn about the upward-opening parabola, how to find roots, and how to interpret function sign changes.", "---", "Stay tuned for more detailed guides on quadratic analysis and inequality solving!"]