Question

How do you write $f\left( x \right)={{x}^{2}}+7x+10$ 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 are going to write down the general equation of vertex and then we are going to use that form to convert the equation to vertex form and also find the (h,k) point. The h point will be taken out by a formula in which we will get the value as negative as both “b” and “a” values are positive i.e. 1 and 7, then we will substitute the value of x i.e. h in the above equation and find out the value of y i.e. k then use those in the standard form to find the final vertex form

Complete step-by-step solution:
In the above question that has been stated where we need to find the vertex from 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 $b=7$ by substituting this we will get the value of x coordinate of the vertex which will be

& \Rightarrow {{x}_{vertex}}=-\dfrac{7}{2\left( 1 \right)} \\\ & \Rightarrow {{x}_{vertex}}=-\dfrac{7}{2} \\\ \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( -\dfrac{7}{2} \right)}^{2}}+7\left( -\dfrac{7}{2} \right)+10 \\\ & \Rightarrow {{y}_{vertex}}=-\dfrac{9}{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( -\dfrac{7}{2} \right) \right)}^{2}}+\left( -\dfrac{9}{4} \right) \\\ & \Rightarrow y={{\left( x+\dfrac{7}{2} \right)}^{2}}-\dfrac{9}{4} \\\ \end{aligned}$$ **The vertex form of the quadratic equation formed is $$y={{\left( x+\dfrac{7}{2} \right)}^{2}}-\dfrac{9}{4}$$.** **Note:** In this type of question there is another way of solving this question, we don’t need to where the vertex of the parabola lies, the quadratic equation which is mentioned in the question when further solved from its expanded form to short form we will get an equation in the form $$4a\left( y-b \right)={{\left( x-c \right)}^{2}}$$ which we can also state it as a parabolic equation, in the equation mentioned above the c and b will become the vertex of the parabola which is h and k respectively in the vertex form and then substitute it to get the answer.