Solve By Factoring: Why Does It Work
Examine the equation below:
If you let a = 3, then logivally b must equal 0. Similarly, if you let b = 10, then a must equal 0.
Now try letting a be some other non-zero number. You should observe that as long as a does not equal 0,b must be equal to zero.
To state the observation more generally, If ab = 0, then either a = 0 or b = 0. This is an important property of zero which we exploit when solving by factoring.
When the example was factored into = 0, this property was applied to determine that either must equal zero, or must equal zero. Therefore, we were able to create two equations and determine two solutions from this observation.
What Does It Mean To Factor An Expression
Factoring an expression means rewriting it as a product of two or more factors.
Really, its another shining moment for the distributive property of multiplication! Were not using it to calculate the product, but we can use it to write an expression as a product.
Factoring is often used when solving quadratic equations because, then, the quadratic expression is rewritten as a product of linear expressions. Then, by remembering the zero product property, we know the solutions of the equation are the solutions of linear equations.
Waitwhat is a quadratic equation?!
Good question! A quadratic equation is an equation in which the variable is raised to the second power.
As a reminder, a quadratic equation written in standard form looks like this:
Why Solve By Factoring
The most fundamental tools for solving equations are addition, subtraction, multiplication, and division. These methods work well for equations like x + 2 = 10 2x and 2 = 0.
But what about equations where the variable carries an exponent, like x2 + 3x = 8x 6? This is where factoring comes in. We will use this equation in the first example.
The Solve by Factoring process will require four major steps:
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Why Is Factoring So Useful
Factoring is used for solving quadratic equations. Some expressions can easily be factored so that the quadratic expression is rewritten as the product of two linear expressions, which we all know how to solve!
The most common factoring is using the square of the sum, the square of the difference, the difference of the squares, and factoring out the common factor.
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