In this comprehensive guide, we will explore two efficient methods for solving quadratic equations on a TI-84 calculator. Whether you prefer using the quadratic formula or leveraging the graphing feature, this article will provide you with step-by-step instructions and practical examples to enhance your understanding of these techniques.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically expressed in the form:

a2 bx c 0

Keywords: Quadratic equation, standard form, coefficients a, b, c

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. Here’s a detailed step-by-step guide on how to use it on a TI-84 calculator.

Step 1: Identify the Coefficients

To begin, rewrite the quadratic equation in the standard form:

a2 bx c 0

Identify the coefficients a, b, and c.

Step 2: Input the Formula into the Calculator

Press the Y button on your TI-84 calculator. Enter the expression for x using the format:
--b ± √(b2 - 4ac) / 2a Use the √ button found under MATH > Numeric for the square root. Calculate the discriminant:
- Discriminant b2 - 4ac Substitute the values of a, b, and c into the formula and press ENTER to find the values of x.

Example

Consider the equation: 22 - 3x - 5 0.

Step 1: Identify the Coefficients

a 2, b 3, and c -5.

Step 2: Calculate the Discriminant

Discriminant 32 - 4(2)(-5) 9 - (-40) 49

Step 3: Find the Values of x

x (-3 ± √49) / 4
(-3 ± 7) / 4
1 or -2.5

The roots of the equation are x1 1 and x2 -2.5.

Method 2: Using the Graphing Feature

The graphing method offers a visual approach to solving quadratic equations. This method involves transforming the equation into a function and finding its x-intercepts.

Step 1: Rearrange the Equation

Rearrange the equation from the form a2 bx c 0

To:

y a2 bx c

Step 2: Input the Equation into the Calculator

Press the Y button. Enter your quadratic equation in the Y1 line. Press the G differentiate (GRAPH) button to see the parabola.

Step 3: Find the Roots of the Equation

After graphing, press the 2nd button then CALC above the TRACE button. Choose 2: Zero to find the x-intercepts (roots). Follow the prompts to select a left bound and a right bound around each root. The calculator will compute and display the x-values where the graph intersects the x-axis.

Example

Consider the equation: 22 - 3x - 5 0.

Step 1: Rearrange the Equation

y 22 - 3x - 5

Step 2: Enter the Equation into the Calculator

Press Y. Enter the equation in the Y1 line. Press G differentiate (GRAPH).

Step 3: Find the Roots of the Equation

Press 2nd then CALC and choose 2: Zero. Select a left bound and a right bound around each root. The calculator will compute and display the roots: x1 1 and x2 -2.5.

Conclusion

Using a TI-84 calculator, you can efficiently solve quadratic equations using either the quadratic formula or the graphing method. These techniques provide accurate and reliable results, making them invaluable tools for both educational and practical applications.