Quadratic Formula

Quadratic Formula

The quadratic formula returns the intercepts of a quadratic equation. The formula is useful when a quadratic equation cannot be easily factored. The result of the formula can be visualized as the points where the quadratic which forms the shape of a parabola intersects with the axis. The generic form of a quadratic equation is given below:

Plugging in the coefficients: , and of the equation into the quadratic formula will produce two results. The symbol can be expanded into two separate equations representing the two solutions:

Quadratic Formula Example

In this example we apply the quadratic formula to the equation . We have set the equation equal to zero because we are interested in where the equation passes through the axis.

Quadratic Formula Example

Steps

  1. Identitify the coefficients , and in the quadratic equation in the form of:

    .

  2. Substitute in for , in for and in for into the quadratic equation.

  3. Evaluate the expression within the square root operator and the expression in the denominator.

  4. Take the square root of and simplify.

  5. Expand symbol into the two solutions.

  6. The two solutions are and .

Derive Quadratic Formula

To derive the quadratic formula, start with the general form of the quadratic equation set equal to zero and solve for the variable . The quadratic formula solves for the x-intercepts of the quadratic equation. The formula is especially useful when the quadratic equation is not easily factored.

Quadratic Formula Expanded

Steps

  1. Divide both sides by .

  2. Subtract from both sides.

  3. Add to both sides to prepare to factor the left side by completing the square.

  4. Factor the left side and simplify the right side by combining the fractions.

  5. Take the square root of both sides.

  6. Subtract from both sides and solve for .

  7. Quadratic Formula.