Today's soundtrack is Petra: Wake-Up Call. I have lots of great memories of this album; I remember listening to it on repeat while playing "Need for Speed III: Hot Pursuit" after getting my homework done for the evening.
This afternoon, I'm learning how to use square roots to solve quadratic equations.
As I learned in the lesson about factoring to solve quadratic equations, the general form of a quadratic equation is ax² + bx + c = 0. We know that a ≠ 0, but either b or c may equal 0.
If either b or c = 0, the form of the equation changes accordingly: for example, if b = 0 in a quadratic equation, the equation will be written as ax² + c = 0. We can then rewrite this equation by moving c to the right side of the equation, giving us ax² = -c. We can then use the square root method to solve the quadratic formula.
As I learned in the lessons about solving radical equations, the square root of a positive number can be either negative or positive; thus, we will always precede it with the symbol ±. If our solution comes out to be the square root of a negative number, we say that the equation has no real roots, as the square root of a negative is not a real number.
If we are given a quadratic equation that can be expressed as (x + p)² = q, we will always be able to solve it by using the square root method. We do this through a process called completing the square, whereby we convert the left side of the equation into a perfect square trinomial, then find the root(s) by finding the square roots of both sides of the equation.
The way to make any quadratic equation into a perfect square is to add the square of 1/2 of the coefficient of x to each side of the equation. If the coefficient of x² is any number other than 1, before we begin, we divide both sides of the equation by that number. Our first step is to factor the left side of the equation so that the left side of the equation is in the form (x ± d)². Next, we combine like terms on the right side of the equation. After that, we find the square root of both sides of the equation, which gives us the root(s) of the equation.
We can use the completing the square method even in situations where factoring doesn't give us a solution!