Desmos for the SAT · Technique 13 of 15
Find a Constant for the Number of Solutions
Drag the constant until the graph shows one crossing, or two, or none.
Some of the trickiest SAT questions hand you an equation with an unknown constant and ask for the value that makes it have exactly one solution, or two, or none, or infinitely many.
The clearest signal is the grey intersection dot. Drag the slider and read the count straight off the graph (see the reference below).
The by-hand way is the discriminant, b squared minus 4ac, and it is a magnet for sign and factor errors. The graphing way skips all of that.
Type the equation with the constant as a letter, turn that letter into a slider, and drag it. Watch how many times the graph crosses the x-axis. The value where you get the count the question wants is your answer.
Nova
Here's a whole family of hard questions that turn easy the second you see them as a graph: the ones asking for the constant that gives 'exactly one solution' or 'no solution.' Instead of doing the discriminant algebra, you drag a slider and literally watch the crossings appear and disappear.
The algebra way vs the Desmos way
This technique sidesteps setting the discriminant b2 - 4ac to zero and solving.
The algebra way
- Write the discriminant:
b^2 - 4ac = 8^2 - 4(1)(c). - Set it to zero for one solution:
64 - 4c = 0. - Solve:
c = 16. Drop the 4 by accident and you getc = 64, which is exactly the trap answer.
The Desmos way
- Type
y = x^2 + 8x + cand add a slider forc. - Drag
cuntil the parabola just touches the x-axis once, landing onc = 16.
The payoff: The slider skips the b2 - 4ac algebra entirely, which is exactly where the "drop the 4" error lives and where the wrong-answer choices are hiding.
See the move, live
That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.
Quick reference
Drag the slider and watch the intersections
| What you see | Number of solutions |
|---|---|
| One grey dot, the line just touches the curve (tangent) | Exactly one |
| The graphs cross at two grey dots | Two |
| The graphs never touch (parallel) | None |
| The two graphs merge into one | Infinitely many |
Step by step
Nova
Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.
Type the equation as y = (the expression), using a letter for the unknown constant, e.g. y = x2 + 8x + c.
Nova: Set it up as y equals the left side. The solutions to the original equation are just wherever this graph hits the x-axis.
Now try it on a real SAT question
In the equation x² + 6x + c = 0, c is a constant, and the equation has exactly one real solution.
What is the value of c?
Pick an answer to get instant coaching from Nova.
"Exactly one real solution" is your signal to set b² - 4ac = 0 and solve for c. Keep that full factor of 4 in there, because dropping it is the classic slip. You never need the actual roots.
Exactly one real solution means the discriminant equals zero: b² - 4ac = 6² - 4(1)(c) = 36 - 4c = 0, so c = 9, choice B. Check: x² + 6x + 9 = (x + 3)², which touches zero only at x = -3.
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
Nova
"Exactly one real solution" is your signal to set b² - 4ac = 0 and solve for c. Keep that full factor of 4 in there, because dropping it is the classic slip. You never need the actual roots.
Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.
The Desmos trap: when to close the calculator
- If the discriminant is quick to set up, like a simple b2 - 4ac = 0, the algebra can be as fast as dragging and you avoid any slider guesswork.
- When the constant that changes the solution count is an ugly fraction or a value the slider cannot land on cleanly, trust b2 - 4ac, not the drag.
- A slider often reveals two values that both work, like k = 4 and k = -4. Before you answer, reread the problem for a constraint like k > 0 and pick the value that fits.
- Sliders default to a range of -10 to 10, so an SAT-scale constant like 25 or 50 sits off the end where you cannot reach it. Click the small numbers at the ends of the slider and widen the range.
Nova
One honest catch: a slider eyeballs it, it does not nail the exact value. If the answer is a clean number like 16 you can confirm it fast, but when the count changes at some ugly fraction, the discriminant is the reliable move. Use the drag to see what is happening, then lock the exact value with the algebra.
Put this move into real practice
"Find a Constant for the Number of Solutions" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.
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