Question

How do you write $f\left( x \right)={{x}^{2}}+14x+45$ in vertex form?

Answer 1

In the above type of question that has been mentioned we need to change the equation in the vertex form for which we will first see what the vertex form for the equation is and what are the things required to make it to vertex form and we will be able to see that we need the vertex point i.e. (h,k) which we will be able to get it from the formula and by substituting those we will be able to find the vertex form of the equation.

Complete step-by-step solution:
In the above question that has been stated where we need to find the vertex form for the given equation for this we are going to first write the general equation of vertex which is:
$y=a{{\left( x-h \right)}^{2}}+k$
Now to write any quadratic equation in vertex form we need to find the vertex coordinates which are h and k. To find those coordinates we will first find the x coordinate of the vertex which can be found with the help of the formula where $x=-\dfrac{b}{2a}$ in this formula x is the x coordinate of the vertex a is the coefficient of the first term of the quadratic equation i.e. ${{x}^{2}}$ and b is the coefficient of the second term of the quadratic equation i.e. x, when we substitute the coefficient values in the formula we will get the x coordinate of the vertex. Now we can find the values of “a'' and “b” from the equation that has been mentioned in the question and we get it as $a=1$ and $b=14$ by substituting this we will get the value of x coordinate of the vertex which will be:

& \Rightarrow {{x}_{vertex}}=-\dfrac{14}{2} \\\ & \Rightarrow {{x}_{vertex}}=-7 \\\ \end{aligned}$$ Now that we have got the x coordinate of the vertex we need to find the y coordinates of the vertex which can be found by substituting the value of x coordinate of the vertex in the quadratic equation mentioned in the question and we will get the y coordinate of the vertex as: $$\begin{aligned} & \Rightarrow {{y}_{vertex}}={{\left( -7 \right)}^{2}}+14\left( -7 \right)+45 \\\ & \Rightarrow {{y}_{vertex}}=-4 \\\ \end{aligned}$$ Now that we know both the coordinates of the vertex we can get the equation of vertex form. As we know that h and k are none other than the x and y coordinate of vertex point we will substitute it in the general equation of vertex form which will result us with the final vertex equation. So the final equation of the vertex is: $$\begin{aligned} & \Rightarrow y=1{{\left( x-\left( -7 \right) \right)}^{2}}+\left( -4 \right) \\\ & \Rightarrow y={{\left( x+7 \right)}^{2}}-4 \\\ \end{aligned}$$ **The vertex form of the quadratic equation formed is $$y={{\left( x+7 \right)}^{2}}-4$$.** **Note:** There is one more way through which we will be able to solve this question, in this method we will first write the general equation of parabola in terms of x$$4a\left( y-b \right)={{\left( x-c \right)}^{2}}$$ and then we are going to use c and b which is the vertex point is required to find the vertex.