Find the Zeros Calculator with Work
Easily find the roots (zeros) of any quadratic equation (ax² + bx + c = 0) and see the step-by-step calculations, including the discriminant and the nature of the roots.
Quadratic Equation Solver (ax² + bx + c = 0)
| Step | Calculation | Value |
|---|---|---|
| 1 | b² | |
| 2 | 4ac | |
| 3 | Discriminant (b² – 4ac) | |
| 4 | √Discriminant | |
| 5 | -b | |
| 6 | 2a | |
| 7 | x1 = [-b + √D] / 2a | |
| 8 | x2 = [-b – √D] / 2a |
What is Finding the Zeros?
Finding the zeros of a function, particularly a quadratic function (which forms a parabola when graphed), means finding the x-values where the function's output (y-value) is equal to zero. These x-values are also known as the roots of the equation or the x-intercepts of the graph. For a quadratic equation in the form ax² + bx + c = 0, the zeros are the values of x that satisfy this equation. Our find the zeros calculator with work helps you find these values easily.
Students of algebra, engineers, physicists, and economists often need to find the zeros of quadratic equations as they model various real-world scenarios like projectile motion, optimization problems, and break-even points. The find the zeros calculator with work is a valuable tool for these calculations.
Common misconceptions include thinking every quadratic equation has two distinct real zeros; sometimes it has one real zero (a repeated root) or two complex zeros.
Quadratic Formula and Mathematical Explanation
The most reliable method to find the zeros of a quadratic equation ax² + bx + c = 0 (where a ≠ 0) is using the quadratic formula:
x = [-b ± √(b² – 4ac)] / 2a
The expression inside the square root, b² – 4ac, is called the discriminant (D). It tells us about the nature of the roots:
- If D > 0, there are two distinct real roots (zeros).
- If D = 0, there is exactly one real root (a repeated root).
- If D < 0, there are two complex conjugate roots (no real zeros).
The find the zeros calculator with work first calculates the discriminant and then applies the full quadratic formula to find the roots, showing each step.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Coefficient of x² | None | Any real number except 0 |
| b | Coefficient of x | None | Any real number |
| c | Constant term | None | Any real number |
| D | Discriminant (b² – 4ac) | None | Any real number |
| x | The zeros/roots | None | Real or Complex numbers |
Practical Examples (Real-World Use Cases)
Example 1: Projectile Motion
A ball is thrown upwards, and its height (h) in meters after time (t) in seconds is given by h(t) = -4.9t² + 19.6t + 1. We want to find when the ball hits the ground (h=0). So, we solve -4.9t² + 19.6t + 1 = 0. Here, a=-4.9, b=19.6, c=1. Using the find the zeros calculator with work:
- a = -4.9, b = 19.6, c = 1
- Discriminant = (19.6)² – 4(-4.9)(1) = 384.16 + 19.6 = 403.76
- t = [-19.6 ± √403.76] / (2 * -4.9) = [-19.6 ± 20.09] / -9.8
- t1 ≈ -0.05 (not valid as time starts at 0), t2 ≈ 4.05 seconds.
The ball hits the ground after approximately 4.05 seconds.
Example 2: Break-Even Analysis
A company's profit (P) from selling x units is given by P(x) = -0.1x² + 50x – 3000. To find the break-even points, we set P(x) = 0: -0.1x² + 50x – 3000 = 0. Using the find the zeros calculator with work with a=-0.1, b=50, c=-3000:
- a = -0.1, b = 50, c = -3000
- Discriminant = (50)² – 4(-0.1)(-3000) = 2500 – 1200 = 1300
- x = [-50 ± √1300] / (2 * -0.1) = [-50 ± 36.06] / -0.2
- x1 ≈ 69.7, x2 ≈ 430.3
The company breaks even when it sells approximately 70 or 430 units.
How to Use This Find the Zeros Calculator with Work
- Enter Coefficients: Input the values for 'a', 'b', and 'c' from your quadratic equation (ax² + bx + c = 0) into the respective fields. 'a' cannot be zero.
- Calculate: Click the "Calculate Zeros" button or simply change the input values. The results will update automatically.
- View Results: The calculator will display:
- The primary result: the zeros (x1 and x2), whether real or complex.
- The discriminant and the nature of the roots.
- A step-by-step breakdown of the quadratic formula application in the "Work Shown" section and the table.
- See the Graph: A simple plot of the parabola y=ax²+bx+c is shown, giving a visual idea of where the zeros might lie (if real).
- Reset or Copy: Use the "Reset" button to go back to default values or "Copy Results" to copy the inputs, results, and work to your clipboard.
The find the zeros calculator with work provides a clear understanding of how the roots are derived. If you are learning about the quadratic formula explained, this tool is very helpful.
Key Factors That Affect Zeros
The zeros of a quadratic equation are entirely determined by the coefficients a, b, and c:
- Value of 'a': If 'a' is close to zero (but not zero), the parabola is very wide, and the roots can be far apart. The sign of 'a' determines if the parabola opens upwards (a>0) or downwards (a<0).
- Value of 'b': The 'b' coefficient shifts the parabola horizontally and influences the axis of symmetry (x = -b/2a). Changes in 'b' move the vertex and thus the roots.
- Value of 'c': The 'c' term is the y-intercept. Changing 'c' shifts the parabola vertically, directly impacting whether the parabola crosses the x-axis and where.
- The Discriminant (b² – 4ac): This combination is crucial. If b² is much larger than 4ac, real roots are likely. If 4ac is larger and positive, complex roots are likely if b² < 4ac. Explore understanding the discriminant for more.
- Relative Magnitudes of a, b, c: The interplay between the magnitudes and signs of a, b, and c determines the location and nature of the zeros.
- Sign of 'a' and 'c': If 'a' and 'c' have opposite signs, 4ac is negative, making b²-4ac positive, guaranteeing two real roots.
Frequently Asked Questions (FAQ)
- Q1: What does it mean if the discriminant is zero?
- A1: If the discriminant (b² – 4ac) is zero, the quadratic equation has exactly one real root (a repeated root). The vertex of the parabola touches the x-axis at exactly one point. Our find the zeros calculator with work will show this.
- Q2: What if 'a' is zero?
- A2: If 'a' is zero, the equation is not quadratic but linear (bx + c = 0), and it has only one root, x = -c/b (if b≠0). This calculator is designed for quadratic equations where a≠0.
- Q3: Can a quadratic equation have no real zeros?
- A3: Yes, if the discriminant is negative (b² – 4ac < 0), the quadratic equation has no real zeros. Instead, it has two complex conjugate roots. The parabola does not intersect the x-axis. See our section on graphing parabolas guide.
- Q4: How does the 'find the zeros calculator with work' handle complex roots?
- A4: When the discriminant is negative, the calculator will show the complex roots in the form p ± qi, where i is the imaginary unit (√-1).
- Q5: What are other methods to find zeros of a quadratic equation?
- A5: Besides the quadratic formula, you can find zeros by factoring the quadratic expression (if it's easily factorable) or by completing the square.
- Q6: Is this calculator the same as a 'solve quadratic equation' tool?
- A6: Yes, "finding the zeros" of ax² + bx + c is the same as "solving the equation" ax² + bx + c = 0. Our polynomial root finder can also be useful for higher degrees.
- Q7: Where is the vertex of the parabola related to the zeros?
- A7: The x-coordinate of the vertex is x = -b/2a, which is exactly halfway between the two real zeros if they exist. Check our vertex of a parabola calculator.
- Q8: Can I use this for any polynomial?
- A8: No, this find the zeros calculator with work is specifically for quadratic polynomials (degree 2). For higher degrees, different methods or more advanced calculators are needed.
Related Tools and Internal Resources
- Quadratic Formula Explained: A detailed guide on how the quadratic formula is derived and used.
- Understanding the Discriminant: Learn more about how the discriminant predicts the nature of the roots.
- Graphing Parabolas Guide: A visual guide to understanding the graphs of quadratic functions.
- Polynomial Root Finder: For finding roots of polynomials of higher degrees.
- Vertex of a Parabola Calculator: Calculate the vertex of your parabola.
- Algebra Basics: Brush up on fundamental algebra concepts.