Question 95·Medium·Equivalent Expressions

After you expand $(x+1)(x-1)$, remember that the entire expression is being subtracted; distribute the negative sign to both terms inside the parentheses.

Once you have a single quadratic expression, group like terms. The value of $b$ is whatever number multiplies $x$ in the final simplified expression.

In Desmos, type y = (3x - 5)^2 - (x + 1)(x - 1) to graph the quadratic expression.

On a new line, type y = a x^2 + b x + c and let Desmos create sliders for a, b, and c.

Adjust the sliders for a, b, and c until the graph of y = a x^2 + b x + c lies exactly on top of the graph of y = (3x - 5)^2 - (x + 1)(x - 1). The value shown for the b slider is the coefficient of $x$ in the equivalent quadratic.

Use FOIL or the square-of-a-binomial formula $(a-b)^2=a^2-2ab+b^2$.

Here, $a=3x$ and $b=5$:

Recognize this as a difference of squares: $(x+1)(x-1)=x^2-1^2$.

So:

The original expression is

Substitute the expanded forms from Steps 1 and 2:

Now distribute the minus sign through the parentheses:

Combine like terms:

• $x^2$ terms: $9x^2 - x^2 = 8x^2$
• $x$ terms: $-30x$ stays as is
• Constant terms: $25 + 1 = 26$

So the expression becomes

This matches the form $ax^2+bx+c$, so $b$ is the coefficient of $x$, which is $-30$.