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"For what value of x": graph both sides, tap the crossing

An original SAT-style practice question on solving an equation, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with the factoring method alongside.

Practice question

x³ − 2x² = 9x − 4

What is the greatest value of x that satisfies the equation above?

⚡ Desmos is the fastest way here Each side becomes its own graph, and every crossing is a solution. No rearranging, no guessing a root, no polynomial division.
Polynomial equationsSolving by graphingHard⏱ ~40 sec

Step by step

1

Graph each side as its own y =

Put y= in front of the left side on one line and the right side on the next. You don't need to move anything across the equals sign.

y=x^{3}-2x^{2}Blue cubic
y=9x-4Red line
−3−2−112356−32−24−16−88162440 y = x³ − 2x² y = 9x − 4 (4, 32) (0.414, −0.272) (−2.414, −25.728)
What you see: an S-shaped curve and a steep line crossing three times. Two crossings sit near the origin and below it; the third is higher up on the right. In a zoomed-in view that one can be off the top of the screen, so zoom out or tap zoom-fit until you can see the curve and line meet on the right.
2

Tap every crossing

Tap where the curve and the line meet. Desmos drops a grey point at each intersection and labels its coordinates, rounded to three decimals.

(−2.414,−25.728)Left crossing
(0.414,−0.272)Middle crossing
(4,32)Right crossing
What you see: three labelled points. Their x-values are the three solutions of the equation.
3

Pick the solution the question asked for

The question wants the greatest value of x, so take the crossing furthest to the right and read its x-coordinate.

4Greatest solution — done
Answer
4

The solutions are x = 4, x ≈ 0.414 and x ≈ −2.414, so the greatest is 4. Check: the left side is 4³ − 2(4²) = 64 − 32 = 32, and the right side is 9(4) − 4 = 32 ✓.

⚠ Common trap

Answering 0.414 because it's the biggest crossing you can see. The x = 4 crossing happens at y = 32, which can sit off the top of a zoomed-in screen, so zoom out before you decide which solution is the greatest. The other slip is answering 32: that's the y-coordinate, the value both sides equal, not x.

The algebra way, if you prefer

Move everything to one side: x³ − 2x² − 9x + 4 = 0. This doesn't factor by grouping, so you have to guess a root. Whole-number roots have to divide the constant 4, so try ±1, ±2 and ±4. x = 4 works: 64 − 32 − 36 + 4 = 0.

Divide out the factor: dividing by (x − 4) leaves x² + 2x − 1, so the equation is (x − 4)(x² + 2x − 1) = 0. The quadratic formula on the second factor gives x = −1 ± √2, which is about 0.414 and −2.414. Both are less than 4, so the greatest solution is 4.

That's correct, but it takes trial and error, one division and the quadratic formula, with a sign to get right at every step. Graphing both sides skips all of it.

More walkthroughs: exponential growth · quadratic roots · systems of linear equations · vertex and maximum value · circle equations · line of best fit · sum of a quadratic's solutions · linear inequalities · absolute value equations · Desmos cheat sheet · all walkthroughs

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