Any time you see an equation, do not solve it. Split it in two. Type the left side as one equation and the right side as another, and read off the point where they meet.
Solve $3x + 7 = 25$.
In Desmos, type $y = 3x + 7$ and $y = 25$. The two lines cross at $x = 6$. That is your answer, and it took about ten seconds.
The real value shows up when equations contain fractions and the algebra then gets messy. Desmos handles fractions just as easily as integers, so the answers come to you clean, quickly and correct.
Solve $\dfrac{2x}{3} - \dfrac{1}{4} = \dfrac{x}{2} + \dfrac{1}{6}$.
By hand you would hunt for a common denominator, clear the fractions and hope nothing slips. In Desmos, type $y = \dfrac{2x}{3} - \dfrac{1}{4}$ and $y = \dfrac{x}{2} + \dfrac{1}{6}$. The two lines cross at $x = 2.5$. No common denominators, no arithmetic slips.
It works just as well on curves.
Solve $x^2 - 4x = 12$.
By hand you would move everything over and factor. In Desmos, type $y = x^2 - 4x$ and $y = 12$. The parabola crosses the line at $x = -2$ and $x = 6$. Both solutions, no factoring, no sign errors.
This is the whole idea. You are not solving the equation, you are letting the two graphs find the answer for you.
Some PSAT questions put a second letter in the equation, an $a$ or a $k$, and ask you to find its value. When you type an equation with a letter Desmos does not recognize, it offers to add a slider. Add it, then drag it until the graph does what the question is asking for.
The line $y = kx + 2$ passes through the point $(4, 14)$. Find $k$.
Type $y = kx + 2$ and add the slider for $k$. Also type the point $(4, 14)$. Now drag the slider until the line runs straight through the point. It lands on $k = 3$.
A whole family of PSAT questions asks how many real solutions an equation has, or how many times a graph meets a line. Once both sides are graphed, the picture tells you at a glance. The rule:
Another thing to keep in mind: solutions, roots and intersections are close synonyms of each other. The PSAT question might use any of these, but they essentially mean the same thing, and the graphical solutions tell the same story.
How many real solutions does $x^2 + 2x + 5 = 0$ have?
Graph $y = x^2 + 2x + 5$ and $y = 0$ (the $x$-axis). The parabola sits entirely above the axis and never touches it. No intersection, no real solutions, no roots. You did not need to calculate a discriminant, because the graph told the story very quickly.
Compare three quick cases. $x^2 - 4 = 0$ crosses the axis twice, at $-2$ and $2$, so two solutions (roots). $x^2 = 0$ touches the axis once, at the origin, so one solution (root). $x^2 + 2x + 5 = 0$ never reaches it, so no solutions (roots). Crosses, touches, misses.
Desmos is built for equations and graphs, and that is where it wins. For a quick piece of arithmetic or a geometry question that is really about an angle or a rule, your pencil is often faster than typing it in. Use the calculator where it saves you time, not on everything. The students who gain the most are the ones who know which questions to hand to it.
On the PSAT, these habits pay twice: they raise your score now, and they carry straight over to the SAT.
Want a plan that builds these habits day by day? The 30 Day PSAT Plan includes two full days of Desmos walkthroughs, 24 videos in all, plus seven full-length practice tests.
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