The function in vertex form is [tex](x-\frac{5}{2} )^{2} -\frac{33}{4}[/tex] and the vertex (h,k) is (-2.5,-8.25).
Completing the Square is a technique which can be used to find maximum or minimum values of quadratic functions. It is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial .
For the given situation,
The function is f(x)=x^2+5x-2.
Plot this function on the graph as shown, this gives the vertex of the function.
The vertex of the function using completing the square method is
⇒ [tex]f(x)=x^2+5x-2[/tex]
Take the coefficient of x and perform the operation,
⇒ [tex](\frac{5}{2}) ^{2} = \frac{25}{4}[/tex]
Now add and subtract this value in the above equation,
⇒ [tex](x^{2} +5x+\frac{25}{4} )-2-\frac{25}{4}[/tex]
⇒ [tex](x-\frac{5}{2} )^{2} -\frac{33}{4}[/tex]
On comparing this equation with the equation in vertex form, we get
⇒ [tex]h=\frac{-5}{2}=-2.5[/tex] and
⇒ [tex]k=-\frac{33}{4}=-8.25[/tex]
⇒ [tex]a=1[/tex]
Hence we can conclude that the function in vertex form is [tex](x-\frac{5}{2} )^{2} -\frac{33}{4}[/tex] and the vertex (h,k) is (-2.5,-8.25).
Learn more about completing the square method here
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