x2 + 8x + 9 = 0

 

Interactive Quadratic Solver

Learn by doing: Solving equations by completing the square.

The Problem

This application will walk you through solving a quadratic equation. Our example equation is:

$$x^2+8x+9=0$$

Step-by-Step Solution

Click the "Next Step" button to reveal each part of the solution. The explanation for each step will appear along with the new equation.

Step 1: Isolate the Constant Term

We move the constant (the term without 'x') to the right side of the equation to begin isolating 'x'.

$$x^2+8x=-9$$

Step 2: Find the 'Completing the Square' Term

To make the left side a perfect square trinomial, we take half of the 'x' coefficient, and square it.

$$(\frac{1}{2}\times 8{)}^2=4^2=16$$

Step 3: Add the Term to Both Sides

We add this new term to both sides to keep the equation balanced. The left side can now be factored as a perfect square.

$$x^2+8x+16=-9+16$$

$$(x+4{)}^2=7$$

Step 4: Take the Square Root

Take the square root of both sides to start solving for 'x'. Remember to include both positive and negative roots.

$$x+4=\pm \sqrt{7}$$

Step 5: Solve for x

Finally, isolate 'x' by subtracting 4 from both sides. This gives us our two solutions.

$$x=-4\pm \sqrt{7}$$

Final Solution

The two roots of the equation are:

$$x=-4+\sqrt{7}\approx -1.354$$

$$x=-4-\sqrt{7}\approx -6.646$$