Calculator for Quadratic Equation - Solver & Graph Tool

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The formula to solve a quadratic equation is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , where $a$ , $b$ , and $c$ are coefficients of the equation $a x^{2} + b x + c = 0$ , and $x$ represents the unknown. Fill in the fields below with the coefficients and press the button to get the unknown x. A graphical solution will also be plotted further down.

Enter the coefficients and click "Calculate Solutions & Graph"

Quadratic Equation Graph

The solutions of the quadratic equation $a x^{2} + b x + c = 0$ may be deduced from the graph of the quadratic function $f (x) = a x^{2} + b x + c$ , which is a parabola.

The graph will be plotted here.

If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept).

If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola.

If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.

— Wikipedia: https://en.wikipedia.org/wiki/Quadratic_equation

How to Obtain the Quadratic Formula

The quadratic formula, $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , is derived from the general form of a quadratic equation, $a x^{2} + b x + c = 0$ , through a process known as completing the square. The derivation can be summarized in the following steps:

Start with the general form: Begin with the quadratic equation in its standard form.
Divide by $a$ : If $a \neq 1$ , divide the entire equation by $a$ to simplify the coefficients. This step makes the coefficient of $x^{2}$ equal to 1, which is necessary for completing the square.
Transpose $c$ : Move the constant term $c$ to the other side of the equation to isolate terms involving $x$ .
Complete the square: Add and subtract the square of half the coefficient of $x$ to the left side of the equation. This step transforms the left side into a perfect square trinomial.
Factor and simplify: Factor the perfect square trinomial on the left side and simplify the right side by combining terms.
Take the square root: Apply the square root to both sides of the equation to solve for $x$ .
Isolate $x$ : Finally, isolate $x$ by moving terms not involving $x$ to the other side and simplifying. This results in the quadratic formula.

The quadratic formula provides the solutions to the quadratic equation by directly substituting the values of $a$ , $b$ , and $c$ into the formula. This method is advantageous because it offers a straightforward way to find the roots of any quadratic equation without needing to factor the equation or complete the square each time.

Why the Name "Quadratic"?

The term "quadratic" comes from the Latin word "quadratus," meaning "square".
This is because the quadratic equation involves the square of the variable (i.e., $x^{2}$ ).
The connection between the operation of squaring in the equation and the geometric square shape leads to the use of the term "quadratic" to describe equations of the form $a x^{2} + b x + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $a \neq 0$ .

Calculator for Quadratic Equation - Solver & Graph Tool

Solve your quadratic equation in just seconds

Solutions:

Quadratic Equation Graph

How to Obtain the Quadratic Formula

Why the Name "Quadratic"?