Question 185·Medium·Equivalent Expressions

Compute $b - 1$ and $a - b$ as separate small steps before expanding any products. What do you get for each of these in terms of $x$?

After you simplify $b - 1$ and $a - b$, expand the product using distribution and then carefully distribute the minus sign in front of $(a - b)$. Are you subtracting both terms inside the parentheses?

In Desmos, type a = 2x - 5 and on the next line type b = x + 2. This lets Desmos treat $a$ and $b$ as expressions depending on $x$.

On a new line, type E(x) = 2a*(b - 1) - (a - b). This defines a function $E(x)$ that represents the original expression after substituting $a$ and $b$.

Add four new lines and enter:

• A(x) = 4x^2 - 7x - 3
• B(x) = 4x^2 - 5x - 12
• C(x) = 4x^2 - x - 17
• D(x) = 2x^2 - 7x - 3 Zoom out if needed so you can see the graphs clearly.

You can either (1) visually compare the graph of $E(x)$ with each of $A(x)$, $B(x)$, $C(x)$, and $D(x)$ to see which one lies exactly on top of $E(x)$ for all $x$, or (2) define difference functions like E(x) - A(x), E(x) - B(x), etc. The choice that gives a graph identical to $E(x)$ (or a horizontal line at $y = 0$ for the difference) is the equivalent expression.

We are given $a = 2x - 5$ and $b = x + 2$.

Start by replacing $a$ and $b$ in the expression $2a(b - 1) - (a - b)$:

Now we just need to simplify step by step.

First simplify $b - 1$ and $a - b$:

• For $b - 1$:

• For $a - b$:

So the whole expression becomes

Now expand $(2x - 5)(x + 1)$:

Multiply this by 2:

Now the expression is

Be careful with the subtraction of $(x - 7)$:

Now combine like terms:

• Combine $-6x$ and $-x$: $-6x - x = -7x$.
• Combine $-10$ and $+7$: $-10 + 7 = -3$.

So the simplified expression is

This matches choice A, so the correct answer is $4x^2 - 7x - 3$.