Question

How do you write the quadratic function in standard form $y=4{{\left( x-1 \right)}^{2}}+5$?

Answer 1

Hint: A quadratic equation is a second-degree polynomial equation having a standard form of $a{{x}^{2}}+bx+c$. Here a and b are coefficients of the variable and c is the constant. Hence a quadratic equation is $f\left( x \right)=y=a{{x}^{2}}+bx+c$. When we substitute an input x in the function, it will give an output y.

Complete Step by Step Solution:
The given equation is $y=4{{\left( x-1 \right)}^{2}}+5$. And we know the standard form of a quadratic equation which is written as
$\Rightarrow y=a{{x}^{2}}+bx+c$
Let us now rearrange the given equation to get the required quadratic form.
$\Rightarrow y=4{{\left( x-1 \right)}^{2}}+5$
We know that ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$. Therefore ${{\left( x-1 \right)}^{2}}$ in the above equation can be written as
$\Rightarrow y=4\left( {{x}^{2}}-2x+1 \right)+5=\left( 4{{x}^{2}}-8x+4 \right)+5$

$\Rightarrow y=4{{x}^{2}}-8x+9$
Where the coefficients $a=4$, $b=-8$ and the constant $c=9$.

The above equation is the standard form of a quadratic function and it is also known as a second-degree polynomial function.

Note:
The quadratic equation is a very important form of an equation that is frequently seen in algebraic problems. Sometimes a problem may require finding a quadratic equation using its roots alone. In that case, we should form an equation containing the roots complemented by a factor. For example, if one of the roots is 3, the corresponding factor should be $\left( x-3 \right)$ which is equal to zero. By multiplying these factors, we can find out the quadratic equation.