Question

How do you find the vertex of $y = 3{x^2} - 6x - 4$?

Answer 1

Hint: Compare the given quadratic equation with the general form given as $y = a{x^2} + bx + c$. Find the respective values of a, b and c. Now, find the discriminant of the given quadratic equation by using the formula $D = {b^2} - 4ac$, where ‘D’ is the notation for the discriminant. Now, use the relation $y = a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}$, where $\left( { - \dfrac{b}{{2a}}, - \dfrac{D}{{4a}}} \right)$ is the vertex.

Complete step-by-step solution:
Here, we have been provided with the quadratic equation $y = 3{x^2} - 6x - 4$ and we are asked to find the vertex.
Now, we are going to use the method of completing the square to solve the question. This method states that if we have a quadratic equation of the form $y = a{x^2} + bx + c$ then its vertex form is written as $y = a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}$, where $\left( { - \dfrac{b}{{2a}}, - \dfrac{D}{{4a}}} \right)$ is the vertex.
So, on comparing the given quadratic expression $y = 3{x^2} - 6x - 4$ with the general form given as $y = a{x^2} + bx + c$, we can conclude that
$ \Rightarrow a = 3,b = - 6,c = - 4$
Applying the formula for the discriminant of a quadratic equation given as ${D^2} = {b^2} - 4ac$, where ‘D’ is the discriminant, we get,
$ \Rightarrow D = {\left( { - 6} \right)^2} - 4 \times 3 \times - 4$
Simplify the terms,
$ \Rightarrow D = 36 + 48$
Add the terms,
$ \Rightarrow D = 84$
Now substitute the values in vertex form,
$ \Rightarrow y = 3{\left( {x + \dfrac{{ - 6}}{{2 \times 3}}} \right)^2} - \dfrac{{84}}{{4 \times 3}}$
Cancel out the common factors,
$ \Rightarrow y = 3{\left( {x - 1} \right)^2} - 7$
Here, the value of $\dfrac{b}{{2a}}$ is $-1$ and $\dfrac{D}{{4a}}$ is $7$.

Hence, the vertex of the quadratic equation is $\left( {1, - 7} \right)$.

Note: You may note that we have derived a general expression for the vertex form. You must remember the result of this vertex form because in the coordinate geometry of the parabola we will be asked to find the vertex of the parabola which can easily be done by this expression. The vertex is given as $\left( { - \dfrac{b}{{2a}}, - \dfrac{D}{{4a}}} \right)$. You may see that we have applied the method of completing the square for the derivation. This method is generally used for the derivation. This method is generally used for finding the roots of a quadratic equation.