Completing the square is a helpful way to find the solutions to quadratic equations.
Let’s break it down step by step:
Start: Begin with a quadratic equation that looks like this:
( ax^2 + bx + c = 0 ).
Rearrange: Move the number ( c ) to the other side:
( ax^2 + bx = -c ).
Complete the Square: For the expression ( x^2 + \frac{b}{a}x ), add ( \left(\frac{b}{2a}\right)^2 ) to both sides. This gives you:
( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -c + \left(\frac{b}{2a}\right)^2 ).
After this step, it looks like this:
( \left( x + \frac{b}{2a} \right)^2 = \text{some value} ).
Now, you can easily find the solutions by taking the square root of both sides and solving for ( x ).
This method not only helps us find the solutions but also gives us a good idea about the vertex (the highest or lowest point) of the parabola created by the quadratic equation.
Completing the square is a helpful way to find the solutions to quadratic equations.
Let’s break it down step by step:
Start: Begin with a quadratic equation that looks like this:
( ax^2 + bx + c = 0 ).
Rearrange: Move the number ( c ) to the other side:
( ax^2 + bx = -c ).
Complete the Square: For the expression ( x^2 + \frac{b}{a}x ), add ( \left(\frac{b}{2a}\right)^2 ) to both sides. This gives you:
( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -c + \left(\frac{b}{2a}\right)^2 ).
After this step, it looks like this:
( \left( x + \frac{b}{2a} \right)^2 = \text{some value} ).
Now, you can easily find the solutions by taking the square root of both sides and solving for ( x ).
This method not only helps us find the solutions but also gives us a good idea about the vertex (the highest or lowest point) of the parabola created by the quadratic equation.