Chapter 1.3, Problem 62PPSE

Solution

i.

To state: The expression $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ for the value $x=3$ .

The resultant value of the expression is 18.

Given information:

The given expression is $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ .

Explanation:

Consider the expression $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ :

  $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$

Substitute the value of x as -3 in the given expression,

  $2\left(2{\left(3\right)}^2-\left(3\right)\right)-3\left({\left(3\right)}^2-\left(3\right)\right)+{\left(3\right)}^2-\left(3\right)$

Simplify powers and multiply,

  $\begin{array}{l}2\left(2\left(9\right)-3\right)-3\left(9-3\right)+9-3\\ 2\left(18-3\right)-3\left(6\right)+9-3\end{array}$

Simplify terms within brackets and multiply,

  $\begin{array}{l}2\left(15\right)-3\left(6\right)+9-3\\ 30-18+9-3\end{array}$

Add and subtract from left to right,

  $12+9-3$

Add and subtract from left to right,

  $21-3=18$

Therefore, the value is 18 for x equal to 3.

ii.

To state: The given expression in part (a) is $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ . Simplify the expression first and the evaluate for $x=3$ .

The resultant answer is 18.

Given information:

The given expression is $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ .

Explanation:

Consider the expression $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ ,

Remove the grouping symbol,

  $4x^2-2x-3x^2+3x+x^2-x$

Use Commutative property,

  $4x^2-3x^2+x^2-2x+3x-x$

Use Distributive property,

  $\left(4-3+1\right)x^2+\left(-2+3-1\right)x$

Combine like terms and simplify,

  $2x^2-\left(0\right)x=2x^2$

Now evaluate $2x^2$ for x equal to 3.

Substitute the value of x as 3,

  $\begin{array}{c}2x^2=2{\left(3\right)}^2\\ =2\left(9\right)\\ =18\end{array}$

iii.

To state: The reason why the values in both parts (a) and (b) should be same.

Because both expressions were equivalent and value of x was same for both the expressions.

Given information:

The given statement says to explain why the values in both parts (a) and (b) should be same.

Explanation:

It can be seen that the resultant values in both parts (a) and (b) are same.

The values of both the parts should be same because the substituted values of x in both the part are same $x=3$ and both of the expressions are equivalent.

In part (a), the value of x is substituted directly in the expression $2\left(2x^2-x\right)-3\left(x^2-x\right)+x^2-x$ but in part (b) the expression was simplified first and then the value of x as 3 is substituted.

In essence both the expressions were equivalent and the substituted value of x is also same for both the parts.