Question 243·Hard·Advanced Math

Equivalent means the expressions have the same value for every $x$, so write the given expression equal to the requested form: $(4x-11)(9x+2)-(5x+13)(7x-6)=ax^2+bx+c$. The target $b$ is the number multiplying $x$; you aren’t looking for a value of $x$.

There are three unknown coefficients, $a$, $b$, and $c$. Three distinct inputs can determine a quadratic, so give Desmos five to fit. Type $x_1=[1...5]$; it shows the list $[1,2,3,4,5]$. The subscript tells Desmos to use these numbers as inputs rather than graph $x$.

Type the equation using $x_1$ in place of every $x$ and $\sim$ in place of $=$. The $\sim$ tells Desmos to fit the unknown coefficients using your list: $(4x_1-11)(9x_1+2)-(5x_1+13)(7x_1-6)\sim ax_1^2+bx_1+c$. Under PARAMETERS, Desmos shows $a=1$, $b=-152$, and $c=56$. So the coefficient of $x$ is $-152$. Grid in -152.

To get an $x$-term, multiply an $x$-term by a constant. Multiplying both $x$-terms makes $x^2$, while multiplying both constants makes no $x$. Multiply the two cross pairs in the first product, then combine: $\begin{aligned}(4x)(2)+(-11)(9x)&=8x-99x\\&=-91x.\end{aligned}$

Use the same cross-pair rule for the second product. Multiply, then combine: $\begin{aligned}(5x)(-6)+(13)(7x)&=-30x+91x\\&=61x.\end{aligned}$

The expression subtracts the whole second product. That minus sign subtracts its $x$-term too. Subtract the coefficients: $\begin{aligned}b&=-91-61\\&=-152.\end{aligned}$ So the coefficient of $x$ is $-152$. Grid in -152.