Question

Find the vertex of $F(x) = {x^2} + 2x - 8 = 0$?

Answer 1

Hint: According to the question we have to determine the vertex of $F(x) = {x^2} + 2x - 8 = 0$. So, first of all we have to compare the given quadratic expression which is $F(x) = {x^2} + 2x - 8 = 0$ with the general form of the expression which is $a{x^2} + bx + c = 0$ to determine the values of $a, b$ and $c$.
Now, to determine the vertex we have to use the formula to determine the value of $x$ which is as mentioned below:

Formula used:
$ \Rightarrow x = - \dfrac{b}{{2a}}..................(A)$
Substitute the value of $x$ in the given quadratic expression which is $F(x) = {x^2} + 2x - 8 = 0$to find the value of $y$.
Now, putting the two numbers together we can determine the coordinate of the vertex.

Complete step by step solution:
First of all we have to compare the given quadratic expression which is $F(x) = {x^2} + 2x - 8 = 0$with the general form of the expression which is $a{x^2} + bx + c = 0$to determine the values of a, b and c which is as mentioned in the solution hint. Hence,
$
   \Rightarrow a = 1, \\
   \Rightarrow b = 2 \\
 $

Now, to determine the vertex we have to use the formula (A) to determine the value of x which is as mentioned in the solution hint. Hence,
$
   \Rightarrow x = \dfrac{{ - (2)}}{{2 \times 1}} \\
   \Rightarrow x = - 1 \\
 $

Now, we have to substitute the value of x in the given quadratic expression which is $F(x) = {x^2} + 2x - 8 = 0$to find the value of y. Hence,
$
   \Rightarrow y = {( - 1)^2} + 2( - 1) - 8 \\
   \Rightarrow y = 1 - 2 - 8 \\
   \Rightarrow y = - 9 \\
 $
On putting the both of the two numbers together we have determined the vertex (-1,-9)

Hence, with the help of the formula (A) we have determined the vertex of the given expression is $( - 1, - 9)$.

Note: Another method:
First of all we have to differentiate the given expression with respect to x. Hence, on differentiating the terms of the expression$F(x) = {x^2} + 2x - 8 = 0$.
$
   \Rightarrow F'(x) = \dfrac{{d{x^2}}}{{dx}} + \dfrac{{d(2x)}}{{dx}} - \dfrac{{d8}}{{dx}} \\
   \Rightarrow F'(x) = 2x + 2 \\
 $
Now, we have to solve the expression after the differentiation with respect to x,
$
   \Rightarrow 2x + 2 = 0 \\
   \Rightarrow x = - 1 \\
 $
Substitute the value of x in the expression to obtain the vertex. Hence,
$
   \Rightarrow F( - 1) = {( - 1)^2} + 2( - 1) - 8 \\
   \Rightarrow F( - 1) = 1 - 2 - 8 \\
   \Rightarrow F( - 1) = - 9 \\
 $
Therefore the vertex of the expression is $( - 1, - 9)$