Question

How do you write \[y = - 2{\left( {x + 1} \right)^2} + 3\] in standard form?

Answer 1

Hint: Here, we will first simplify the entire equation by using algebraic identity. Then we will solve the equation formed on the right side using basic mathematical operation and write it in the standard form of second degree polynomial to get the required answer.

Formula Used:
Algebraic Identity- \[\left( {a + b} \right) = {a^2} + 2ab + {b^2}\], where, \[a,b = \] any real number.

Complete step by step solution:
The standard form of two degree polynomial is \[y = a{x^2} + bx + c\].
The polynomial that is to be written in standard form is
\[y = - 2{\left( {x + 1} \right)^2} + 3\]…………………………....\[\left( 1 \right)\]
We will solve the bracket term by using algebraic identity given below, \[\left( {a + b} \right) = {a^2} + 2ab + {b^2}\].
Using above formula in equation \[\left( 1 \right)\], we get
\[ \Rightarrow y = - 2\left( {{x^2} + 2 \times x \times 1 + {1^2}} \right) + 3\]
Simplifying the equation, we get
$\Rightarrow y=-2{{x}^{2}}-4x-2+3$
Subtracting the like terms, we get
\[ \Rightarrow y = - 2{x^2} - 4x + 1\]

So we get the standard form of \[y = - 2{\left( {x + 1} \right)^2} + 3\] as \[y = - 2{x^2} - 4x + 1\].

Additional Information:
Polynomials are formed by two terms one is ‘poly’ which means ’many’ and next is ‘nomial’ which means terms in this case. Polynomials are easy to work with as we know that when we add polynomials we get a polynomial also if we multiply polynomials we get a polynomial. So no matter how much polynomials are being used in the equation the result will always be a polynomial.

Note:
In the standard form, the terms are written from highest degree to lowest degree depending upon the degree of the polynomial. If the solution we got was not in the highest to lowest order we would have changed it and written accordingly. A constant term having no variable always comes in the last and the term with the highest power variable always comes in the first.