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. To factor a quadratic equation, the standard form is more beneficial. The standard form of a quadratic equation is ax^2 + bx + c = 0. This form allows you to easily identify the values of a, b, and c, which are necessary for factoring. Factoring involves finding two binomials that multiply together to give the original quadratic equation. By factoring a quadratic equation in standard form, you can easily identify the roots or x-intercepts.
b. To graph a parabola, the vertex form is more beneficial. The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. By using the vertex form, you can easily identify the coordinates of the vertex, which is essential to graph the parabola accurately. Additionally, the vertex form also allows you to determine whether the parabola opens upwards or downwards based on the value of a.
c. To identify the vertex (minimum or maximum) of a parabola, the vertex form is more beneficial. As mentioned earlier, the vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) represents the vertex. By directly reading the values of h and k from the equation, you can easily determine the coordinates of the vertex, which provides information about the minimum or maximum point on the parabola.
d. To solve a quadratic equation using the quadratic formula, the standard form is more beneficial. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. The quadratic formula requires the quadratic equation to be in the standard form (ax^2 + bx + c = 0) to directly substitute the values of a, b, and c into the formula. By organizing the quadratic equation in standard form, you can effortlessly apply the quadratic formula to find the solutions for x.
X squared +7×+6=0
a standard b vertex
(c) obviously the vertex form
(d) standard form, so you have the needed a,b,c coefficients.
What do you think for (a) and (b)?
Try doing them and see what happens.