Quadratic Formula Calculator

A Quadratic Formula Calculator is a powerful tool that helps solve quadratic equations, providing accurate solutions for the variables in an equation of the form ax² + bx + c = 0.

What Does Quadratic Formula Calculator Do?

The Quadratic Formula Calculator is designed to help solve quadratic equations by applying the well-known quadratic formula. It is particularly useful for students, engineers, and anyone working with polynomial equations. The tool quickly provides the roots of the equation, offering solutions for both real and complex numbers.

Quadratic Formula Calculation Formula

The quadratic formula is used to solve equations of the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

Where:

• a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0.
• The discriminant (b² – 4ac) helps determine the nature of the roots:
  • If positive, there are two real roots.
  • If zero, there is one real root.
  • If negative, the roots are complex.

Quadratic Formula Calculator Examples

Example 1: Solving a Quadratic Equation with Real Roots

• Equation: 2x² + 4x – 6 = 0
• Step 1: Identify the coefficients:
  • a = 2, b = 4, c = -6
• Step 2: Substitute into the quadratic formula: x = (-4 ± √(4² - 4(2)(-6))) / 2(2)
• Step 3: Calculate the discriminant (b² – 4ac): b² - 4ac = 4² - 4(2)(-6) = 16 + 48 = 64
• Step 4: Calculate the square root of the discriminant: √64 = 8
• Step 5: Substitute back into the formula: x = (-4 ± 8) / 4
• Step 6: Solve for the two roots:
  • First root: x = (-4 + 8) / 4 = 4 / 4 = 1
  • Second root: x = (-4 – 8) / 4 = -12 / 4 = -3
• Result: The roots of the equation are x = 1 and x = -3.

Example 2: Solving a Quadratic Equation with Complex Roots

• Equation: x² + 2x + 5 = 0
• Step 1: Identify the coefficients:
  • a = 1, b = 2, c = 5
• Step 2: Substitute into the quadratic formula: x = (-2 ± √(2² - 4(1)(5))) / 2(1)
• Step 3: Calculate the discriminant (b² – 4ac): b² - 4ac = 2² - 4(1)(5) = 4 - 20 = -16
• Step 4: Since the discriminant is negative, the roots are complex. Calculate the square root of -16: √-16 = 4i
• Step 5: Substitute back into the formula: x = (-2 ± 4i) / 2
• Step 6: Solve for the two roots:
  • First root: x = (-2 + 4i) / 2 = -1 + 2i
  • Second root: x = (-2 – 4i) / 2 = -1 – 2i
• Result: The roots of the equation are x = -1 + 2i and x = -1 – 2i.

The Quadratic Formula Calculator provides an efficient way to solve quadratic equations and helps you find both real and complex roots, making it a vital tool for anyone working with algebraic expressions.