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Quadratic roots: graph it, tap the x-intercepts

An original SAT-style practice question on the roots of a quadratic, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with factoring and the quadratic formula alongside.

Practice question

In the xy-plane, the graph of y = 2x² + 5x − 42 intersects the x-axis at the points (a, 0) and (b, 0), where a < b. What is the value of b?

⚡ Desmos is the fastest way here The x-intercepts are the roots, and Desmos marks both the moment you tap the curve — no factoring, no quadratic formula.
QuadraticsRoots / x-interceptsMedium⏱ ~25 sec

Step by step

1

Graph the equation as written

Type it exactly as given. The bottom of this parabola sits around y = −45, so if it runs off the screen, tap the zoom-fit button (the magnifying glass) to bring the whole curve into view.

y=2x^{2}+5x-42A parabola opening upward appears
−7−5−4−3−2−1125−40−30−20−101020 (−6, 0) (3.5, 0)
What you see: a U-shaped curve that dips well below the x-axis and crosses it twice — once on the left of the y-axis and once on the right.
2

Tap the curve to show the x-intercepts

Tap the parabola. Desmos drops grey points where it crosses the x-axis and labels each one with its coordinates. Those x-values are the roots.

(−6,0)Left crossing
(3.5,0)Right crossing
What you see: two labelled points on the x-axis, at (−6, 0) and (3.5, 0).
3

Pick the root the question asked for

The question names the points so that a < b, which makes b the larger x-value — the crossing on the right.

3.5The larger root — done
Answer
3.5  (or 7/2)

The graph crosses the x-axis at x = −6 and x = 3.5, so a = −6 and b = 3.5. Check: 2(3.5)² + 5(3.5) − 42 = 24.5 + 17.5 − 42 = 0 ✓ and 2(−6)² + 5(−6) − 42 = 72 − 30 − 42 = 0 ✓.

⚠ Common trap

Answering −6 — that's a, the smaller root. Read which one the question wants (a < b, the positive one, the larger one) before you answer. If you factor by hand, watch the signs too: the factor (x + 6) gives the root x = −6, not 6, and (2x − 7) gives x = 7/2, not 7.

The algebra way, if you prefer

Factoring: look for two numbers that multiply to 2 · (−42) = −84 and add to 5: that's 12 and −7. Split the middle term and group: 2x² + 12x − 7x − 42 = 2x(x + 6) − 7(x + 6) = (2x − 7)(x + 6). Set each factor to zero: x = 7/2 or x = −6. The larger one is 7/2.

Quadratic formula: the discriminant is 5² − 4(2)(−42) = 25 + 336 = 361 = 19², so x = (−5 ± 19) / 4, giving 14/4 = 3.5 or −24/4 = −6.

Both work, but they take longer and give you more places to slip a sign. When the numbers don't factor nicely, graphing wins by even more.

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

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