Quadratic Formula Calculator

Quadratic Formula Calculator is a Quadratic Equation Solver to solve a second-order polynomial equation such as a ² + b + c = 0 for x where a ≠ 0, using the  quadratic formula and completing the square method.

Quadratic Formula Method

The quadratic formula =(− ±√( ²−4 ))/2  is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2) ax2 + bx + c = 0.

In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation.

Δ = b² – 4ac

• If the discriminant is zero, then there is exactly one real root

= (− /2 )

• If the discriminant is positive, then there are two distinct roots

= (− + √Δ)/2   and = (− – √Δ)/2

• If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots. i is the imaginary unit.

= − /2 + i(√-Δ)/2a   and =− /2 – i(√-Δ)/2a

Completing the Square

Completing the square method is where we take a Quadratic Equation like a ² + b + c = 0  and turn it into a( + d)² + e = 0. This method uses the algebraic formula (a+b)² = a² + b² + 2ab

Steps to convert a ² + b + c = 0 into a( + d)² + e = 0.

1. Divide each side by a, the coefficient of the squared term.
2. Subtract the constant term c/a from both sides. Add the square of one-half of b/a, the coefficient of x, to both sides.
3. This “completes the square”, converting the left side into a perfect square.