Completing the Square Calculator

The “Completing the Square Calculator” is designed to help you transform any quadratic equation of the form $ax^2 + bx + c = 0$ into a perfect square trinomial. This method is useful for solving quadratic equations and understanding the geometric properties of parabolas, such as their vertex and axis of symmetry.

Key Features

1. Input Fields: Enter the coefficients $a$, $b$, and $c$ from the quadratic equation.

   • $a$: The coefficient of $x^2$.
   • $b$: The coefficient of $x$.
   • $c$: The constant term.

2. Calculate Button: After entering the values of $a$, $b$, and $c$, click the “Calculate” button to perform the completing the square process.

3. Clear Button: If you want to clear the inputs and start over, click the “Clear” button.

4. Answer Section: After clicking the “Calculate” button, the solution will be displayed below, including all the steps and the final result.

Formulas for Completing the Square

Step-by-Step Process

1. Start with the quadratic equation:

   $$a{x}^2+bx+c=0$$

   If $a \neq 1$, divide the entire equation by $a$ to simplify:

   $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$

2. Move the constant $\frac{c}{a}$ to the right-hand side:

   $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

3. Add the square of half the coefficient of $x$: Take $\left( \frac{b}{2a} \right)^2$ and add it to both sides:

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

4. Factor the left-hand side: The left-hand side now becomes a perfect square trinomial:

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

5. Solve for $x$: Finally, solve for $x$ by taking the square root of both sides and isolating $x$:

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

6. The final solutions are:

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

Example 1: Solving $x^2 + 2x + 1 = 0$

1. The quadratic equation is $x^2 + 2x + 1 = 0$, where $a = 1$, $b = 2$, and $c = 1$.

2. The steps are as follows:

   $$x^2+2x=-1$$

   Add $\left( \frac{2}{2} \right)^2 = 1$ to both sides:

   $$x^2+2x+1=0$$

   Factor the left side:

   $$(x+1{)}^2=0$$

   Solving for $x$, we get:

   $$x=-1$$

   Therefore, the solution is $x = -1$.

Example 2: Solving $2x^2 + 8x – 10 = 0$

1. The quadratic equation is $2x^2 + 8x – 10 = 0$, where $a = 2$, $b = 8$, and $c = -10$.

2. First, divide the equation by 2 to simplify:

   $$x^2+4x=5$$

   Add $\left( \frac{4}{2} \right)^2 = 4$ to both sides:

   $$x^2+4x+4=9$$

   Factor the left side:

   $$(x+2{)}^2=9$$

   Solve for $x$:

   $$x+2=\pm 3$$

   Therefore, the solutions are:

   $$x=1\text{or}x=-5$$