How to Solve Quadratic Equations on a TI-84 Calculator

Introduction

Quadratic equations can be solved with factoring, completing the square, the quadratic formula, or graphing. A TI-84 can help you estimate roots, check exact algebra, and understand what the solutions mean visually.

The calculator is especially useful when the roots are not easy whole numbers. It can also help you catch mistakes after solving by hand. If your class requires exact answers, use the TI-84 as a checking tool rather than the only method.

Step-by-step graphing method

1. Move all terms to one side so the equation equals zero.
2. Press Y=.
3. Enter the quadratic expression in Y1.
4. Press GRAPH.
5. Look for x-intercepts.
6. Use trace or zero-finding features to estimate the roots.

This method works because the solutions of `ax^2 + bx + c = 0` are the x-values where the graph of `y = ax^2 + bx + c` crosses the x-axis. At those points, y equals zero.

Practical example

For `x^2 - 5x + 6 = 0`, graph `y = x^2 - 5x + 6`. The graph crosses the x-axis at `2` and `3`, matching the factored form `(x - 2)(x - 3)`.

This is a good example because the calculator and algebra agree cleanly. If you factor the expression first, the graph confirms your answer. If you graph first, the intercepts suggest how the expression factors.

Using a table to check roots

The table can confirm roots without relying only on the picture. Enter the expression in Y1, open TABLE, and look for x-values where y equals `0`.

For `x^2 - 5x + 6`, the table shows `0` at x-values 2 and 3 if your step includes those values. If the roots are decimals, use a smaller table step near the crossing point.

Using the quadratic formula as a check

The quadratic formula is:

`x = (-b +/- sqrt(b^2 - 4ac)) / (2a)`

For `x^2 - 5x + 6 = 0`, the coefficients are `a = 1`, `b = -5`, and `c = 6`. The formula gives the same roots, `2` and `3`.

When the graph gives a decimal answer, the quadratic formula can tell you whether the exact answer includes a square root. This is important in classes that want exact solutions instead of rounded estimates.

What if there are no x-intercepts?

Some quadratics do not cross the x-axis. That means there are no real roots. The graph may sit entirely above or below the x-axis.

Before deciding that there are no real roots, check the window. If the graph is off screen, you may simply be looking in the wrong place. Use ZOOM or adjust WINDOW to make sure the vertex and x-axis are visible.

What if there is only one x-intercept?

If a quadratic touches the x-axis at the vertex and turns around, it has one repeated real root. Algebraically, this means the discriminant is zero. Graphically, the parabola is tangent to the x-axis.

Common mistakes

• Graphing both sides without understanding what intersection means.
• Forgetting to set the equation equal to zero.
• Trusting a graph estimate when the problem asks for exact roots.
• Using a window that hides one intercept.
• Forgetting parentheses when entering negative coefficients.
• Rounding roots too early.

FAQ

Can a TI-84 solve every quadratic exactly?

It can help check solutions, but exact answers may require algebra.

What if the graph does not cross the x-axis?

The quadratic may have no real roots, or your window may hide them.

Can I solve by graphing two sides separately?

Yes, but then you are finding intersections between two graphs. For basic quadratic equations, moving everything to one side and finding zeros is usually cleaner.

Should I use the TI-84 or the quadratic formula?

Use both when possible. The calculator helps you check and visualize. The quadratic formula gives exact answers when required.

What related article should I read?

Read how to graph on a TI-84 and practice on the homepage calculator.