This find roots calculator uses the quadratic formula to solve for the two roots ($x_1$ and $x_2$) of a second-degree polynomial equation: $ax^2 + bx + c = 0$. Input the coefficients A, B, and C to get real or complex solutions instantly.
find roots calculator
find roots calculator Formula
The roots ($x$) of the quadratic equation $ax^2 + bx + c = 0$ are found using the Quadratic Formula:
$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$Formula Source: Wikipedia – Quadratic Formula | Wolfram MathWorld
Variables Explained
The calculator requires three coefficients to define the polynomial:
- Coefficient A: The number multiplying the squared term ($x^2$). This cannot be zero.
- Coefficient B: The number multiplying the linear term ($x$).
- Coefficient C: The constant term (the y-intercept).
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What is find roots calculator?
A “find roots calculator” is a mathematical tool used to determine the values of the variable that satisfy an equation, meaning the values that make the equation true (equal to zero). In the context of a quadratic equation ($ax^2 + bx + c = 0$), these roots are also known as the zeros of the function, or the x-intercepts of the corresponding parabola.
The ability to find roots is fundamental in various fields, including physics (trajectory analysis), engineering (circuit design), and economics (optimization problems). The nature of the roots depends entirely on the discriminant ($\Delta = b^2 – 4ac$): they can be two distinct real numbers, a single repeated real number, or two complex conjugate numbers.
This calculator simplifies the process by automating the quadratic formula, eliminating manual calculation errors and instantly determining the roots, regardless of whether they are real or complex.
How to Calculate find roots calculator (Example)
Let’s find the roots for the equation: $2x^2 + 8x – 10 = 0$ (A=2, B=8, C=-10):
- Identify Coefficients: $a=2$, $b=8$, $c=-10$.
- Calculate the Discriminant ($\Delta$): $$\Delta = b^2 – 4ac = (8)^2 – 4(2)(-10) = 64 – (-80) = 144$$
- Apply the Quadratic Formula: $$x = \frac{-8 \pm \sqrt{144}}{2(2)} = \frac{-8 \pm 12}{4}$$
- Solve for $x_1$ (using +): $$x_1 = \frac{-8 + 12}{4} = \frac{4}{4} = 1$$
- Solve for $x_2$ (using -): $$x_2 = \frac{-8 – 12}{4} = \frac{-20}{4} = -5$$
- Result: The roots are 1 and -5.
Frequently Asked Questions (FAQ)
Is it possible to have zero roots for a quadratic equation?
No. A quadratic equation (where $A \ne 0$) will always have exactly two roots. These roots might be real and distinct, real and repeated, or complex conjugates, but there will always be two solutions (counting multiplicity).
What does it mean when the roots are complex?
Complex roots occur when the discriminant ($b^2 – 4ac$) is negative. It means the graph of the parabola ($y = ax^2 + bx + c$) never crosses the x-axis, so there are no real number solutions. The roots involve the imaginary unit $i$ ($i = \sqrt{-1}$).
Can I use this calculator for linear equations?
A linear equation is an equation where $A=0$. If you set $A=0$, the calculator will show an error because the quadratic formula requires division by $2a$. For linear equations ($bx + c = 0$), the root is simply $x = -c/b$.
What is the discriminant’s role?
The discriminant ($\Delta = b^2 – 4ac$) determines the nature of the roots without fully calculating them. $\Delta>0$ means two real roots, $\Delta=0$ means one real root, and $\Delta<0$ means two complex roots.