Question 60·SAT Desmos Regression Practice·Advanced Math

0 / 68 Questions completed

(2x^2+ax+b)(3x^2+cx-5)=6x^4+7x^3-x^2-19x-20

The equation above is true for all $x$ , where $a$ , $b$ , and $c$ are constants. Which choice gives the value of $a-c$ ?

The words “true for all $x$ ” signal an identity: both sides are the same polynomial, not an equation where you solve for $x$ . Use a Desmos identity regression at several inputs to find the unknown constants, or match coefficients by hand. One matching input isn't enough to determine all the constants. Once you have them, calculate the requested difference rather than stopping at one constant.

Hints

Hint 1
An identity holds at every value of $x$ . That means you can use several distinct inputs to fit the unknown constants; one input alone could let many different sets of constants produce the same value.
Hint 2
In a Desmos regression, write the inputs as a list called $x_1$ . Replace each $x$ in the given equation with $x_1$ , and replace $=$ with $\sim$ so Desmos fits $a$ , $b$ , and $c$ .
Hint 3
The target is $a-c$ , not one of the fitted constants. Read $a$ and $c$ under PARAMETERS, then subtract in the order the question gives. How does the subtraction work if $c$ is negative?

Step-by-step

Approach 1: Fit the identity in Desmos
Step 1
Find the three constants
An identity holds at every $x$ . If both sides agree for every $x$ , their coefficients must match. Type $x_1=[-2,-1,0,1,2]$ to give Desmos five distinct inputs. Then type the given equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . The multiple inputs keep one accidental match from standing in for the identity. Under PARAMETERS, Desmos shows $a=3$ , $b=4$ , and $c=-1$ .
Step 2
Calculate the requested difference
Type $a-c$ on the next line. Desmos prints $4$ . Here $c=-1$ , so subtracting $c$ means subtracting a negative: $a-c=3-(-1)=4$ . The value of $a-c$ is $4$ . Choice D.

Approach 2: Match coefficients by hand
Step 1
Match the constant terms
A constant term has no $x$ . On the left it comes from $b(-5)$ ; on the right it is $-20$ . Since the identity holds for every $x$ , match them: $-5b=-20$ . Divide by $-5$ : $b=4$ .
Step 2
Match the $x^3$ terms
The $x^3$ terms come from multiplying $(ax)(3x^2)$ and $(2x^2)(cx)$ : $(ax)(3x^2)+(2x^2)(cx)=(3a+2c)x^3$ . A coefficient is the number multiplying a power of $x$ . Match this coefficient to the $7$ multiplying $x^3$ on the right: $3a+2c=7$ .
Step 3
Match the $x$ terms
Use $b=4$ . The $x$ terms come from $(ax)(-5)$ and $4(cx)$ : $(ax)(-5)+4(cx)=(-5a+4c)x$ . Match that coefficient to the $-19$ multiplying $x$ on the right: $-5a+4c=-19$ .
Step 4
Eliminate $c$ to find $a$
The second equation has $4c$ , so double $3a+2c=7$ to give the equations matching $c$ terms: $6a+4c=14$ . Subtract $-5a+4c=-19$ ; the $4c$ terms cancel: $11a=33$ . Divide by $11$ : $a=3$ .
Step 5
Find $c$
Put $a=3$ into the equation $3a+2c=7$ : $3(3)+2c=7$ . Subtract $9$ : $2c=-2$ . Divide by $2$ : $c=-1$ . Keep the negative sign for the final subtraction.
Step 6
Subtract in the requested order
The question asks for $a-c$ , not $c-a$ . Substitute the values: $a-c=3-(-1)=4$ . So the value of $a-c$ is $4$ . Choice D.

Question 60·SAT Desmos Regression Practice·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this