#8 · Test one value per unknown
The question. In the given equation, a, b, and c are constants. What is the value of a + b + c?
By hand. The two sides are the same expression written in different ways, so make the left side look like the right side.
Clear the fractions. The common denominator is (x − 1)(x + 3), which multiplies out to x² + 2x − 3: the right side's denominator. Multiply every term by it. Each fraction's own denominator cancels, which leaves two products on the left and only the top on the right.
Expand each product, then combine like terms.
Match it to the right side.
So a = 2, b = −7, and c = −10.
That makes a + b + c = 2 − 7 − 10 = −15.
The Desmos shortcut: a regression
The graphing calculator built into Bluebook is Desmos, and Desmos can find a, b, and c for you. Type the equation as a regression: a tilde (~) in place of the equals sign, and x₁ in place of x. (Plain x won't work: Desmos says x may not be used as a regression parameter.) On the next line, type a + b + c to see the sum.
Two more lists to avoid
- Avoid pairs that add to zero, like 2 and −2. x₁ = [−2, 0, 2] looks like three tests, but Desmos gives −9.75. Here a and b sit on x³ and x, so −2 gives the same numbers as 2 with the signs flipped, and Desmos learns nothing new from it. The safe habit: use values like 0, 2, and 3.
- Keep the values small. Huge values like a million push Desmos's rounding too far: [1000000, 2000000, 3000000] gives about −5. Small whole numbers are safest.
This question type is on the test: the College Board lists equivalent expressions under Advanced Math, and Desmos courses such as Test Prep Wizards teach the regression for it. The divide-by-zero check is our own addition.
The takeaway: make both sides look alike. In Desmos, test one value per unknown, and skip any value that makes a denominator zero.