Algebra: Chapter 6, Lesson 8, p 286
Solving Equations by Factoring
First, we need the principle of ZERO PRODUCTS. If `a * b = 0`, then we must have either `a = 0` or `b = 0` or both a and b = 0.
If we have an equation with 0 on 1 side and a factorization on the other, we can solve the equation by finding the values that make the factors 0.
Example: `(5x + 1)(x − 7) = 0`. This means that either `(5x + 1) = 0` or `(x − 7) =0`. Solving the first equation, we have `x = −1/5`. Solving the second equation, we also have `x = 7`.
The 2nd part of the lesson is Factoring and Solving. Here we can:
- Get products on 1 side of the equation using the addition property
- Factor the expression on the other side of the equation
- Set each factor equal to 0
- Solve each equation.
As an example, we have `x^2 − 8x = −16` or `x^2 − 8x + 16 = 0`. We can factor the left side to be `(x − 4)(x − 4) = 0`, so that the solution turns out to be `x = 4`.
Remember, with these quadratic equations, we can have AT MOST 2 solutions. We can have 0, 1 or 2 solutions with quadratics, the 2 in the exponent tells us how many AT MOST solutions we will find.
As usual, purplemath.com also has some nice explanations.
Math-8: Chapter 9, Lesson 4, p 444
Using Proportions
You can use proportions to solve problems that relate to ratios. A PROPORTION is a statement of equality of 2 or more ratios. Consider `a/b = c/d`. You can CRISS-CROSS and come up by multiplying with `ad = bc`. The products are called CROSS PRODUCTS of a proportion.
We can also go backwards, if `ad = bc`, then `a/b = c/d`.
Scaling drawings and models, cooking recipes and other things fall into proportions too.
As an example, 4 cups of flour yield 64 cookies. If we have 6 cups of flour, how many cookies can we make?
We set up a proportion as follows:
`4/64 = 6/x` or CRISS-CROSSING, we end up with `4x = 6 * 64` or `4x = 384` or `x = 384/4` or finally, `x = 96`