Keywords:
Quadratic equation, vertex shape, parabola
For this case we have to rewrite the given quadratic equation, in the form of vertex, for this, we must take into account that a quadratic equation of the form [tex]ax ^ 2 + bx + c = 0[/tex], can be rewritten in the form of vertex as: [tex]y = a (x-h) ^ 2 + k.[/tex] Vertice is the lowest or highest point of the parabola. The vertex is given by: [tex](h, k)[/tex]. So, let: [tex]f (x) = 4x ^ 2 + 48x + 10[/tex], to find the equation in the form of vertex, we follow the steps below:
Step 1:
We take the common factor to the first two terms of the equation:
[tex]f (x) = 4 (x^2 + 12x) + 10[/tex]
Step 2:
We work square:
We divide the coefficient of the term [tex]"12x"[/tex] by 2 and its result is squared, that is:
[tex](\frac {12} {2}) ^ 2 = 36[/tex]
So, we have:
[tex]f (x) = 4 (x^2 + 12x + 36-36) + 10[/tex]
Step 3:
We simplify:
[tex]f (x) = 4 (x^2 + 12x + 36) + 10- (4 * 36)[/tex]
Step 4:
We factor:
[tex]f (x) = 4 (x + 6) ^ 2-134[/tex]
Thus, [tex]h = -6\ and\ k = -134[/tex]
Answer:
The equation in the form of vertex is: [tex]f (x) = 4 (x + 6) ^ 2-134[/tex], and the vertex is [tex](h, k) = (- 6, -134)[/tex]
