Find the Roots Calculator With Steps (Quadratic)
Quadratic Equation Roots Calculator (ax²+bx+c=0)
Enter the coefficients a, b, and c of your quadratic equation to find its roots using the quadratic formula, with step-by-step calculations shown.
What is Finding the Roots of a Quadratic Equation?
Finding the roots of a quadratic equation (an equation of the form ax² + bx + c = 0, where a ≠ 0) means finding the values of 'x' for which the equation holds true. These values of 'x' are also known as the "solutions" or "zeros" of the equation. Graphically, the roots are the x-intercepts of the parabola represented by y = ax² + bx + c, i.e., the points where the parabola crosses the x-axis.
This find the roots calculator with steps helps you determine these values by applying the quadratic formula and showing the intermediate calculations.
Anyone studying algebra, or professionals in fields like engineering, physics, and finance who deal with quadratic relationships, would use a tool or method to find these roots. It's fundamental in understanding the behavior of quadratic functions.
Common misconceptions include thinking every quadratic equation has two distinct real roots (it can have one real root or two complex roots) or that the formula is always easy to apply without care (especially with negative numbers under the square root).
The Quadratic Formula and Mathematical Explanation
A quadratic equation is given by:
ax² + bx + c = 0 (where a ≠ 0)
To find the roots (the values of x), we use the quadratic formula, which is derived by completing the square:
x = [-b ± √(b² – 4ac)] / 2a
The term inside the square root, b² – 4ac, is called the discriminant (Δ). The discriminant tells us about the nature of the roots:
- If Δ > 0, there are two distinct real roots.
- If Δ = 0, there is exactly one real root (a repeated root).
- If Δ < 0, there are two complex conjugate roots (no real roots).
Our find the roots calculator with steps first calculates the discriminant and then the roots based on its value.
Variables in the Quadratic Formula:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Coefficient of x² | Dimensionless | Any real number except 0 |
| b | Coefficient of x | Dimensionless | Any real number |
| c | Constant term | Dimensionless | Any real number |
| Δ (b²-4ac) | Discriminant | Dimensionless | Any real number |
| x | Roots of the equation | Dimensionless | Real or complex numbers |
Practical Examples (Real-World Use Cases)
Example 1: Projectile Motion
The height 'h' of an object thrown upwards after time 't' can be modeled by h(t) = -gt²/2 + v₀t + h₀, where g is gravity, v₀ is initial velocity, and h₀ is initial height. To find when the object hits the ground (h=0), we solve -gt²/2 + v₀t + h₀ = 0. If g=9.8, v₀=20, h₀=1, we solve -4.9t² + 20t + 1 = 0. Using the find the roots calculator with steps with a=-4.9, b=20, c=1, we find the times 't' when the object is at height 0 (one positive time for hitting the ground after launch).
Inputs: a = -4.9, b = 20, c = 1
Discriminant ≈ 419.6
Roots (t) ≈ 4.13 seconds and -0.05 seconds. The positive root is the time it takes to hit the ground.
Example 2: Area Optimization
Suppose you have 40 meters of fencing to enclose a rectangular area, and you want the area to be 96 square meters. If the sides are x and (20-x), the area is x(20-x) = 20x – x². Setting this to 96 gives x² – 20x + 96 = 0. We can use the find the roots calculator with steps with a=1, b=-20, c=96 to find the possible dimensions.
Inputs: a = 1, b = -20, c = 96
Discriminant = 16
Roots (x) = 8 and 12. So the dimensions could be 8m by 12m.
How to Use This Find the Roots Calculator With Steps
- Enter Coefficients: Input the values for 'a', 'b', and 'c' from your quadratic equation (ax² + bx + c = 0) into the respective fields. Ensure 'a' is not zero.
- View Results: The calculator will automatically update and show the primary result (the roots x1 and x2, or a message if roots are not real) and intermediate values like the discriminant.
- Examine Steps: The "Calculation Steps" table will show how the discriminant and roots were calculated using the quadratic formula.
- See the Graph: The chart visually represents the parabola y = ax² + bx + c and highlights the x-intercepts (the real roots).
- Reset or Copy: Use the "Reset" button to clear the inputs to their default values or "Copy Results" to copy the main results and steps.
The results will clearly state if the roots are real and distinct, real and repeated, or complex. If they are complex, this calculator will indicate that there are no real roots.
Key Factors That Affect the Roots
The nature and values of the roots of a quadratic equation are determined entirely by the coefficients a, b, and c.
- Coefficient 'a': Affects the width and direction of the parabola. If 'a' is large, the parabola is narrow; if small, it's wide. If 'a' is positive, it opens upwards; if negative, downwards. Changing 'a' changes the roots' values and the vertex position.
- Coefficient 'b': Influences the position of the axis of symmetry (x = -b/2a) and thus the location of the vertex and roots along the x-axis.
- Coefficient 'c': This is the y-intercept of the parabola (where x=0). It shifts the parabola up or down, directly impacting the y-coordinate of the vertex and the values of the roots.
- The Discriminant (b² – 4ac): This is the most crucial factor determining the *nature* of the roots.
- If b² – 4ac > 0, there are two distinct real roots (parabola crosses x-axis at two points).
- If b² – 4ac = 0, there is exactly one real root (parabola touches x-axis at one point – the vertex).
- If b² – 4ac < 0, there are no real roots; the roots are complex conjugates (parabola does not intersect the x-axis).
- Ratio b/a and c/a: The sum of the roots is -b/a, and the product of the roots is c/a. Changes in these ratios directly shift or scale the roots.
- Sign of 'a' and 'c': If 'a' and 'c' have opposite signs, 'ac' is negative, making -4ac positive, increasing the discriminant and guaranteeing real roots (as b² is always non-negative).
Our find the roots calculator with steps clearly shows the discriminant and then uses it to determine the roots.
Frequently Asked Questions (FAQ)
A quadratic equation is a second-order polynomial equation in a single variable x, with the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.
If 'a' were zero, the ax² term would vanish, and the equation would become bx + c = 0, which is a linear equation, not quadratic.
The discriminant (b² – 4ac) tells us the nature of the roots: positive means two distinct real roots, zero means one real root (repeated), and negative means two complex roots (no real roots).
Complex roots occur when the discriminant is negative. They involve the imaginary unit 'i' (where i² = -1) and are not represented on the real number line (the x-axis in the graph). This find the roots calculator with steps focuses on real roots.
No, according to the fundamental theorem of algebra, a polynomial of degree 'n' has exactly 'n' roots (counting multiplicity and complex roots). A quadratic is degree 2, so it has exactly two roots.
The vertex is the point where the parabola reaches its minimum (if a>0) or maximum (if a<0) value. Its x-coordinate is -b/(2a).
If the discriminant is negative, the calculator will indicate that there are no real roots and will not display numerical values for x1 and x2 in the primary result for real roots, though it shows the discriminant.
Yes, but you first need to rearrange your equation into the standard form to identify the correct values of a, b, and c before using the find the roots calculator with steps.
Related Tools and Internal Resources
- Quadratic Formula Explained: A detailed explanation of how the quadratic formula is derived and used. Our quadratic formula calculator is also there.
- Discriminant and Nature of Roots: Learn more about how the discriminant determines the types of roots. We have a discriminant calculator too.
- Graphing Quadratic Functions: Understand how to graph parabolas and their relationship to roots.
- Algebra Calculators: Explore other calculators for various algebraic problems, including an equation solver online.
- Math Solvers: A collection of tools to help solve different mathematical problems.
- Calculus Tools: Calculators related to calculus concepts.