Question

How do you write $y=\left( x-4 \right)\left( x-2 \right)$ in standard form?

Answer 1

Hint: At first, we open up the brackets by multiplying the terms within them using the distributive property to get $y=\left( {{x}^{2}}-2x \right)-\left( 4x-8 \right)$ . After that, we club the like terms together and simplify the expression to get $y={{x}^{2}}-6x+8$ . The standard quadratic expression is $a{{x}^{2}}+bx+c$ so, we arrange the derived expression in this fashion, which will be our final answer.

Complete step by step solution:
The given equation that we have at our disposal is,
$y=\left( x-4 \right)\left( x-2 \right)$
In order to express it in some random form, we first need to multiply the terms inside the brackets. For this, we apply the distributive property. According to the distributive property, an expression of the form $\left( a+b \right)\left( c+d \right)$ can be written as $a\left( c+d \right)+b\left( c+d \right)$ . Transforming or modifying the given equation accordingly, the given equation thus becomes,
$\Rightarrow y=x\left( x-2 \right)-4\left( x-2 \right)$
We can observe that the equation has not been transformed or simplified entirely to the simplest form as there are brackets still present. So, we need to apply the distributive property again this time. But here, instead of handling multiplication of two brackets, we handle one bracket this time. Thus, an expression of the form $a\left( b+c \right)$ under the distributive property becomes $ab+ac$ . We now apply this property to the above equation and the above equation thus becomes,
$\Rightarrow y=\left( {{x}^{2}}-2x \right)-\left( 4x-8 \right)$
Simplifying the above equation by adding the like terms, we get,
$\Rightarrow y={{x}^{2}}-6x+8$
Now, the above equation can be compared to the standard equation of a quadratic equation. The standard quadratic equation is $y=a{{x}^{2}}+bx+c$ where a and b are called the coefficients of the ${{x}^{2}}$ and x terms respectively.
Therefore, we can conclude that the given expression can be standardised to $y={{x}^{2}}-6x+8$.

Note: The distributive property may seem easy and it really is. But the major problem associated with the distributive property execution is the mistake with signs. So, we should be careful here. Also, we should ensure that no line terms have remained unadded or not.