Question

What is the vertex form of $y={{x}^{2}}-5x-3$ ?

Answer 1

Here in this question we have been asked to give the vertex form of $y={{x}^{2}}-5x-3$ for answering this question we will first simplify the given expression and write it in the vertex form that is here we can say that the given expression represents a parabola and the vertex form equation means that the expression should be in terms of the vertex of the given parabola.

Complete step by step answer:
Now considering the question we have been asked to give the vertex form of $y={{x}^{2}}-5x-3$ .
From the basic concepts we know that a quadratic equation generally represents a parabola and the vertex form equation means that the expression should be in terms of the vertex of the given parabola.
Now we can say that the given expression represents a parabola.
Now by simplifying the given expression by making only $x$ terms on the right hand side we will have $\Rightarrow y+3={{x}^{2}}-5x$ .
Now we will further simplify this expression
$\begin{aligned} & \Rightarrow y+3={{x}^{2}}-5x+{{\left( \dfrac{5}{2} \right)}^{2}}-{{\left( \dfrac{5}{2} \right)}^{2}} \\\ & \Rightarrow y+3+\dfrac{25}{4}={{\left( x-\dfrac{5}{2} \right)}^{2}} \\\ & \Rightarrow y+\dfrac{37}{4}={{\left( x-\dfrac{5}{2} \right)}^{2}} \\\ \end{aligned}$ .
Here we can say that the vertex of the given parabola will be given as $\left( \dfrac{5}{2},\dfrac{-37}{4} \right)$ because at this the whole equation tends to be zero.

Therefore we can conclude that the vertex form of $y={{x}^{2}}-5x-3$ will be given as $y+\dfrac{37}{4}={{\left( x-\dfrac{5}{2} \right)}^{2}}$ .

Note: This is a very simple and easy question and very few mistakes are possible in this type of question. Here below we can see the graph of the given expression: