To solve a quadratic on a TI-84 Plus CE, press APPS, open PlySmlt2, choose Polynomial Root Finder, set the order to 2, and enter a, b, and c. Press SOLVE to see both roots. Without the app, graph the equation and use 2nd TRACE, 2 for zero at each place the graph crosses the x-axis.
How to solve a quadratic on a TI-84 with PlySmlt2
PlySmlt2 comes preloaded on the TI-84 Plus CE. Its menu bar sits at the bottom of the screen, and you choose each item with the key right below it. You can practice the app's menu bar keys on the free TI-84 calculator online, which uses the same keypad.
- Press APPS.
- Scroll to PlySmlt2, press ENTER, then press any key at the title screen.
- Press 1 for Polynomial Root Finder.
- Set the order to 2, then press the key below NEXT on the menu bar.
- Type a, b, and c into the three boxes, pressing ENTER after each.
- Press the key below SOLVE to show x1 and x2.
If the app is missing, use the built in equation solver instead. Press MATH, scroll to the bottom of the list, and choose Solver. After eqn: 0=, type the quadratic using the X,T,θ,n key for x, then press ENTER.
Type a guess for X, then press ALPHA ENTER to solve. The Solver returns one root, the one nearest your guess. Enter a new guess on the other side of the vertex to get the second root.
What the discriminant tells you
The discriminant is D = b² − 4ac, the part under the square root in the quadratic formula. Its sign tells you what kind of roots you will get before you solve.
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0 | Two different real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis at the vertex |
| D < 0 | Two complex roots | Never touches the x-axis |
When D is a perfect square, such as 64, the roots are rational and the quadratic factors over the integers. That is why the calculator above shows a factored form for some equations and not others.
Worked example with real and complex roots
Real roots: 2x² − 4x − 6 = 0
Here a = 2, b = −4, and c = −6. The discriminant is (−4)² − 4(2)(−6) = 16 + 48 = 64, and its square root is 8.
Plug into the formula: x = (4 ± 8) ÷ 4. The plus sign gives 12 ÷ 4 = 3, and the minus sign gives −4 ÷ 4 = −1. So the roots are x = 3 and x = −1, and the equation factors as 2(x + 1)(x − 3).
The vertex sits at x = −b ÷ 2a = 1. Plug 1 back in and y = 2 − 4 − 6 = −8, so the vertex is (1, −8).
Complex roots: x² + 2x + 5 = 0
Now a = 1, b = 2, and c = 5. The discriminant is 4 − 20 = −16, which is negative. The roots are x = (−2 ± 4i) ÷ 2, which simplifies to −1 + 2i and −1 − 2i.
To get these on the home screen, press MODE, move down to REAL, select a+bi, and press ENTER. Then type (−2+√(−16))÷2 and the TI-84 returns −1+2i. In PlySmlt2, set the app's own MODE screen to a+bi before you solve.
Graphing the parabola to check
A quick graph confirms your roots. Real roots are where the parabola crosses the x-axis, so you should see it cross at −1 and 3 in the first example.
- Press Y= and type 2X²−4X−6 next to Y1.
- Press ZOOM, then 6 for ZStandard.
- Press 2nd TRACE, then 2 for zero.
- Move left of a crossing and press ENTER, move right of it and press ENTER, then press ENTER once more.
For the vertex, use 2nd TRACE, 3 for minimum when a is positive, or 4 for maximum when a is negative. You can do all of this in the online graphing calculator without a handheld.
Common mistakes
- Not setting the equation equal to zero. Rewrite x² = 3x + 4 as x² − 3x − 4 = 0 before reading off a, b, and c.
- Dropping the sign of b. If the equation has −4x, then b is −4, and −b in the formula becomes +4.
- Typing −16 with the subtraction key. Use the (−) key for negative numbers on a TI-84.
- Leaving the calculator in REAL mode for a negative discriminant. Switch MODE to a+bi to see complex roots. The TI-84 error codes table explains the message you get otherwise.
FAQ
Does the TI-84 have a quadratic formula program?
Not as a program, but the TI-84 Plus CE ships with the PlySmlt2 app, which solves quadratics directly. Open it from APPS and choose Polynomial Root Finder. You can also write a short program in PRGM, but the app and the built in Solver already cover every case a class needs.
How do you download PolySmlt2 on a TI-84?
The app's real name is PlySmlt2, and it is already installed on most TI-84 Plus CE calculators. Check APPS first. If it was deleted, download it free from the Texas Instruments education website and send it to the calculator with the TI Connect CE software and a USB cable.
How do you solve a quadratic equation on a TI-84 without a program?
Graph it and find the zeros. Type the quadratic in Y1, press ZOOM 6, then press 2nd TRACE and 2 for zero. Set a left bound, a right bound, and a guess around each crossing. You can also use the Solver at the bottom of the MATH menu.
How do you get imaginary answers on a TI-84?
Press MODE, move down to the row that starts with REAL, select a+bi, and press ENTER. Now square roots of negative numbers return answers with i instead of an error. For example, √(−16) gives 4i. In PlySmlt2, choose a+bi on the app's MODE screen instead.
What does ERR DOMAIN mean when solving a quadratic?
ERR:DOMAIN means a function received a value outside what it accepts. With quadratics, a square root of a negative number in REAL mode usually shows ERR:NONREAL ANS instead. Check your signs and parentheses, or switch MODE to a+bi if the discriminant is negative.
How do you find the vertex on a TI-84?
Graph the quadratic, then press 2nd TRACE and choose 3 for minimum if it opens up or 4 for maximum if it opens down. Set a left bound, a right bound, and a guess around the turn. The screen shows the vertex as X and Y values.