Quadratic Formula Calculator

Enter coefficients for ax² + bx + c = 0:

The quadratic formula is a fundamental tool in algebra used to solve quadratic equations. A quadratic equation is a second-degree polynomial equation of the form:

\[ ax^2 + bx + c = 0 \]

Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents the variable. The quadratic formula provides the solutions (or roots) of this equation, which are the values of \( x \) that satisfy the equation.

The Quadratic Formula

The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula calculates the roots of the quadratic equation by taking into account the coefficients \( a \), \( b \), and \( c \). The term under the square root, \( b^2 - 4ac \), is called the discriminant. The discriminant determines the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root (a repeated root).
  • If the discriminant is negative, there are two complex roots.

How to Calculate Using the Quadratic Formula

Let’s break down the steps to solve a quadratic equation using the quadratic formula:

  1. Identify the coefficients: For the equation \( ax^2 + bx + c = 0 \), identify the values of \( a \), \( b \), and \( c \).
  2. Calculate the discriminant: Compute \( D = b^2 - 4ac \).
  3. Determine the nature of the roots: Based on the value of \( D \), decide whether the roots are real or complex.
  4. Apply the quadratic formula: Substitute \( a \), \( b \), and \( D \) into the formula to find the roots.

Example

Solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \).

Step 1: Identify the coefficients: \[ a = 2, \quad b = -4, \quad c = -6 \]

Step 2: Calculate the discriminant: \[ D = b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 \]

Step 3: Since \( D > 0 \), there are two distinct real roots.

Step 4: Apply the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4} \] This gives two solutions: \[ x_1 = \frac{4 + 8}{4} = 3 \] \[ x_2 = \frac{4 - 8}{4} = -1 \]

Solution: The roots are \( x = 3 \) and \( x = -1 \).

Note

If the discriminant is negative, the quadratic formula will yield complex roots. For example, if \( D = -9 \), the roots will be: \[ x = \frac{-b \pm \sqrt{-9}}{2a} = \frac{-b \pm 3i}{2a} \] where \( i \) is the imaginary unit (\( i^2 = -1 \)).

Finally

The quadratic formula is a powerful and straightforward method for solving quadratic equations. By understanding the discriminant and following the steps outlined above, you can easily determine the roots of any quadratic equation. Whether the roots are real or complex, the quadratic formula provides a clear and systematic way to find them.

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