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A quadratic equation can be written in vertex form or in standard form. Sometimes one form is more beneficial than the other. Identify which form would be more helpful if you needed to do each task listed below and explain why.
a. Factor the equation.
b. Graph the parabola.
c. Identify the vertex, minimum, or maximum of the parabola.
d. Solve the equation using the quadratic formula.
a. Factor the equation: The standard form of a quadratic equation, ax^2 + bx + c = 0, is more helpful for factoring the equation. This is because the quadratic equation can be factored by finding the product of two binomials, (x + p)(x + q), where p and q are the roots of the equation. In standard form, it is easier to identify the coefficients a, b, and c, which are necessary for factoring.
b. Graph the parabola: The vertex form of a quadratic equation, a(x - h)^2 + k, is more helpful for graphing the parabola. This is because the vertex form directly provides the coordinates of the vertex of the parabola, (h, k). By using the vertex form, one can easily sketch the graph by shifting the basic parabolic shape horizontally or vertically.
c. Identify the vertex, minimum, or maximum of the parabola: The vertex form of a quadratic equation, a(x - h)^2 + k, is more helpful for identifying the vertex, minimum, or maximum of the parabola. The vertex of the parabola is represented by the values (h, k) in the vertex form. Additionally, the coefficient "a" in the vertex form determines whether the parabola opens upward (minimum) or downward (maximum).
d. Solve the equation using the quadratic formula: The standard form of a quadratic equation, ax^2 + bx + c = 0, is more helpful for solving the equation using the quadratic formula. The quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), requires identification of the coefficients a, b, and c, which are readily available in the standard form. By substituting these coefficients into the quadratic formula, one can easily solve for the roots of the equation.