Quadratic Formula Calculator

What is the Quadratic Formula?

The quadratic formula is a mathematical tool used to solve quadratic equations of the form:

$$ax^2 + bx + c = 0$$

The solution to this equation is given by:

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

Here:

• $a$, $b$, and $c$ are coefficients of the quadratic equation,
• $b^2 - 4ac$ is called the discriminant, determining the nature of the roots.

Nature of Roots Based on the Discriminant:

1. Positive Discriminant ($b^2 - 4ac > 0$): Two distinct real roots.
2. Zero Discriminant ($b^2 - 4ac = 0$): One real root (repeated root).
3. Negative Discriminant ($b^2 - 4ac < 0$): Two complex roots.
   
   

Derivation of the Quadratic Formula

Step 0: Start form the quadratic formula, or from the general quadratic equation:

$$ax^2 + bx + c = 0$$ Step 1: Normalize the equation

Divide through by $a$ (assuming $a \neq 0$) to simplify the equation:

$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ Step 2: Move the constant term to the other side

Rearrange the equation so that the constant term is isolated:

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$ Step 3: Complete the square

To complete the square, add and subtract $\left(\frac{b}{2a}\right)^2$ on the left-hand side:

$$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$$

Simplify the left-hand side as a perfect square trinomial:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}$$ Step 4: Combine terms on the right-hand side

The fractions on the right-hand side can be combined under a common denominator:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$ Step 5: Take the square root of both sides

Apply the square root to both sides, remembering to consider both the positive and negative roots:

$$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ Step 6: Solve for $x$

Isolate $x$ by subtracting $\frac{b}{2a}$ from both sides:

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

Combine the terms into a single fraction:

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

Example: Solve

$2x^2 - 4x - 6 = 0$

1. $a = 2$, $b = -4$, $c = -6$.
2. Substitute into the formula: $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)}$$ $$x = \frac{4 \pm \sqrt{16 + 48}}{4}$$ $$x = \frac{4 \pm \sqrt{64}}{4}$$ $$x = \frac{4 \pm 8}{4}$$
3. Simplify: $$x_1 = \frac{4 + 8}{4} = 3, \quad x_2 = \frac{4 - 8}{4} = -1$$

Sources:

Keedy, M. L., & Bittinger, M. L. (1982). Algebra and Trigonometry: A Functions Approach. Addison Wesley Publishing Company.