Question

How do you write the quadratic \[y = - 1.8{x^2} - 4.32x + 4.896\] in vertex form?

Answer 1

Hint: Here, we will first take a value common from all the terms of the equation. Then we will use the completing the square method to simplify the equation. Then we will use the general equation of vertex form to write the obtained equation in vertex form.

Complete step-by-step answer:
The quadratic equation given to us is \[y = - 1.8{x^2} - 4.32x + 4.896\].
Now, taking \[ - 1.8\] common in right hand side from first two values we get,
\[ \Rightarrow y = \left( { - 1.8} \right)\left( {{x^2} + 2.4x} \right) + 4.896\]
Next, we will complete the square of the bracket term having variable \[x\].
So, we will add and subtract \[{\left( {1.2} \right)^2}\] in bracket having variable \[x\] to get
\[y = \left( { - 1.8} \right)\left( {{x^2} + 2.4x + {{\left( {1.2} \right)}^2} - {{\left( {1.2} \right)}^2}} \right) + 4.896\]
Simplifying the equation further, we get
\[\begin{array}{l} \Rightarrow y = \left( { - 1.8} \right)\left( {{x^2} + 2.4x + {{\left( {1.2} \right)}^2}} \right) - \left( { - 1.8} \right) \times {\left( {1.2} \right)^2} + 4.896\\ \Rightarrow y = \left( { - 1.8} \right)\left( {{x^2} + 2.4x + {{\left( {1.2} \right)}^2}} \right) + 2.592 + 4.896\end{array}\]
Adding the like terms, we get
\[ \Rightarrow y = \left( { - 1.8} \right)\left( {{x^2} + 2.4x + {{\left( {1.2} \right)}^2}} \right)7.488\]
Now using the identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], we get
\[ \Rightarrow y = \left( { - 1.8} \right){\left( {x + 1.2} \right)^2} + 7.488\]……………………………………..\[\left( 1 \right)\]
We know that the vertex form of a quadratic equation is \[y = m{\left( {x - a} \right)^2} + b\].
So, we will simplify the equation \[\left( 1 \right)\] as general vertex form of equation to get,
\[ \Rightarrow y = \left( { - 1.8} \right){\left( {x - \left( { - 1.2} \right)} \right)^2} + 7.488\]

So, vertex form of \[y = - 1.8{x^2} - 4.32x + 4.896\] is \[y = \left( { - 1.8} \right){\left( {x - \left( { - 1.2} \right)} \right)^2} + 7.488\] with vertex \[\left( { - 1.2,7.488} \right)\].

Note:
The vertex form of a quadratic equation is \[y = m{\left( {x - a} \right)^2} + b\] where the slope of the equation is \[m\] and \[a,b\] is the vertex of \[x,y\] respectively. The vertex form is an alternative way to write the equation of a parabola. We need to keep in mind that in vertex form the value of \[a\] is subtracted and the value of \[b\] is added. The common mistake we can make in vertex form is of the sign one should always double check the positive and negative sign in this type of form and don’t get confused in what is the value of the vertex. Vertex form is very useful when we are dealing with parabola figure questions.