Hi Class!
One possible quadratic equation in the form ax^2 + bx + c = 0 where a > 1 is 2x^2 – 5x + 2 = 0.
To find the number of solutions based on the discriminant, we can use the formula b^2 – 4ac. In this case, b = -5, a = 2, and c = 2, so the discriminant is (-5)^2 – 4(2)(2) = 1. Since the discriminant is positive and not equal to zero, this quadratic equation has two distinct real solutions.
To find the specific solutions, we can use the quadratic formula or graph the equation.
Using the quadratic formula, we have: x = (-b ? ?(b^2 – 4ac)) / 2a Substituting the values of a, b, and c, we get: x = (5 ? ?(5^2 – 4(2)(2))) / 4 Simplifying this expression, we get: x = (5 ? ?17) / 4 Therefore, the solutions are approximately 1.28 and 0.22.
Alternatively, we can graph the equation and find the points where the graph intersects the x-axis. By factoring or completing the square, we can determine the vertex of the parabola and the axis of symmetry, which can also help in graphing the equation. In this case, the vertex is (5/4, -1/8) and the axis of symmetry is x = 5/4. The graph intersects the x-axis at approximately x = 1.28 and x = 0.22.
I chose the quadratic formula and graphing methods because they are two of the most common and reliable methods for finding the solutions of a quadratic equation. Factoring and completing the square can also be effective, but may not always work for every quadratic equation. The square root property can only be used for quadratic equations with a single squared term and a constant term, so it is not applicable in this case.
References
Kremer, M. (2018). Using the quadratic formula to solve equations. Mathematics Teacher, 111(6), 438-443. Retrieved from https://www.jstor.org/stable/26671019Links to an external site.
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