Quadratic Equation Calculator

A quadratic equation is defined as an equation in the form:

Where:

a, b, c are constants,

x is the variable.

The key feature of a quadratic equation is that the variable x is raised to the second power.

Finding the roots of a quadratic equation means discovering all the values of x that satisfy the equation.

The discriminant is an important indicator used to determine the number and type of roots for the quadratic equation ax²+bx+c = 0. It is represented by the symbol (D) and calculated using the formula: D = b² − 4ac.

Where:

a, b, c are the coefficients of the quadratic equation ax²+bx+c = 0.

The value of the discriminant D can take on three possible scenarios:

1. If D>0 , the equation has two distinct real roots.

2. If D=0 , there is exactly one real root.

3. If D<0 , there are no real roots, but the equation has complex roots.

By evaluating the discriminant, one can determine the presence and number of roots of a quadratic equation without directly calculating the roots themselves. Therefore, understanding the discriminant is essential when analyzing quadratic equations.

Quadratic Equation without Real Roots (D < 0): If the discriminant is less than zero, the equation has no real roots. Graphically, this means the parabola does not intersect the x -axis, and the solutions will consist of complex numbers.

Quadratic Equation with One Real Root (D = 0): When the discriminant is zero, the equation has precisely one real root, which will be the same for both methods of solving the quadratic equation. Graphically, this indicates that the parabola is tangent to the x -axis.

Quadratic Equation with Two Distinct Real Roots (D > 0): If the discriminant is greater than zero, the equation has two different real roots. Graphically, this implies that the parabola intersects the x -axis at two distinct points.

There are several types of quadratic equations based on the coefficients a, b, c and the values on the right side of the equation. Here are some examples:

Standard Quadratic Equation: ax²+bx+c = 0.

• where a, b, с can be any numbers.

Equation of the Form ax² = 0

• Here, the equation simplifies to ax² = 0.
  The solution to this equation will be x = 0 for any non-zero value of a .

Equation of the Form ax²+bx+c = 0.

• If the coefficient b is zero ( b=0 ), the equation simplifies to ax² + c = 0. The solution will depend on the signs and values of c and a.

Equation of the Form ax²+bx+c = 0.

• If there is no constant term, i.e., c = 0, the equation takes the form ax² + bx = 0. The solution will depend on the values of a and b.

Complete Square Equations:

• These are in the form (x + a)² = b or (x - a)² = b, where a and b are known numbers.

Mixed Types of Equations:

• These can include combinations of the above forms, such as ax² + c = bx.

Once you've found the roots of a quadratic equation, you can verify their accuracy by substituting them back into the original equation. If both sides of the equation remain equal, then your solution is correct!