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.

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.