Day 148 – April 23

Algebra: Chapter 13, Lesson 3, page 586.

Solving quadratics by completing the squares

We can use the technique of completing the square to solve quadratic equations. Recall that the addition property allows us to add a number to both sides of the equation. To complete a square, take ${\displaystyle \frac{1}{2}}$ of the x-coefficient, square it and add it to both sides of the quadratic equation.

For example:

${\displaystyle x^2–4x–7=0}$ becomes ${\displaystyle x^2–4x=7}$ by adding 7 to both sides.

Now, take ${\displaystyle \frac{1}{2}}$ of the x coefficient (the ${\displaystyle -4}$ in ${\displaystyle -4x}$). ${\displaystyle \frac{1}{2}}$ of the ${\displaystyle -4}$ is ${\displaystyle -2}$. Now square that, ${\displaystyle -2^2=4}$ and

${\displaystyle x^2–4x+4=7+4}$ by adding 4 to both sides to complete the square.

REMEMBER: ${\displaystyle x^2–4x+4}$ factored becomes ${\displaystyle {(x–2)}^2}$

${\displaystyle {(x–2)}^2=11}$ or by square rooting each side to ${\displaystyle x–2=\pm \sqrt{11}}$ so that

${\displaystyle x=2\pm \sqrt{11}}$ or ${\displaystyle 2+\sqrt{11}}$ and ${\displaystyle 2–\sqrt{11}}$ as the final solutions.

Purplemath explains it too, just click here for step-by-step instructions.