How to Solve Quadratic Equations on a TI-84 Plus CE (Step by Step)

Every equation of the form ax² + bx + c = 0 (with a ≠ 0) has exactly two solutions, given by the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

The three coefficients are read directly off the equation — for 2x² - 5x + 3 = 0, that’s a = 2, b = -5, c = 3.

The discriminant decides what kind of answer you get

The expression under the square root, b² - 4ac, is called the discriminant, and its sign tells you what the two roots will look like before you finish the calculation:

• Positive discriminant — two distinct real roots. The parabola crosses the x-axis at two different points.
• Zero discriminant — one repeated real root. The parabola just touches the x-axis at its vertex.
• Negative discriminant — two complex roots, a conjugate pair like p + qi and p − qi. The parabola never touches the x-axis at all — it’s entirely above or entirely below it.

That third case is not an error condition. x² + 1 = 0 has no real solution because no real number squares to -1 — but it has two perfectly valid complex ones, i and -i, and the quadratic formula produces them the same mechanical way it produces real roots: √(-4) becomes 2i and the rest of the formula proceeds exactly as normal.

Worked example

For x² - 4x + 5 = 0: a = 1, b = -4, c = 5, so the discriminant is (-4)² - 4(1)(5) = 16 - 20 = -4. Negative, so expect a complex conjugate pair. Plugging in:

x = (4 ± √-4) / 2 = (4 ± 2i) / 2 = 2 ± i

The two roots are 2 + i and 2 - i.

Graphing the parabola instead of solving algebraically

Every quadratic is also a parabola, and graphing it answers the same question a different way. On a physical TI-84 Plus CE: type the equation into the Y= editor (Y1 = X² - 4X + 5), press GRAPH, then use 2ND + TRACE (the CALC menu) to read specific features off the curve without eyeballing pixels:

• 2: zero finds where the parabola crosses the x-axis — the real roots, if there are any. If the discriminant is negative, the curve never touches the axis and zero won’t find a crossing, which is itself confirmation you’re looking at a complex-root case.
• 3: minimum or 4: maximum (whichever applies, depending on whether the parabola opens up or down) finds the vertex.

Full walkthrough of the Y= editor, window sizing, and Trace/Zoom in general — not specific to parabolas — in How to Graph on a TI-84.

Try it yourself

The TI-84 Calculator’s Tools → Solver tab has a dedicated quadratic solver — enter a, b, and c and it returns both roots directly, real or complex, without you needing to evaluate the discriminant by hand first. It uses the same underlying complex-number engine as the calculator’s Complex mode, so the two always agree. To see the parabola itself instead, switch to Graph mode, enter the same equation in the Functions tab, and use the same zoom/pan/trace workflow as the physical device.