Welcome to factoring quadratics! Factoring means breaking down a quadratic expression like ax squared plus bx plus c into a product of two simpler expressions. Let's start with a basic example: x squared plus 5x plus 6. We need to find two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so our factored form is x plus 2 times x plus 3.
When the coefficient of x squared is not 1, we use the AC method. For 2x squared plus 7x plus 3, we multiply a times c which is 2 times 3 equals 6. We need two numbers that multiply to 6 and add to 7. Those are 6 and 1. We split the middle term: 2x squared plus 6x plus 1x plus 3. Then factor by grouping: 2x times x plus 3, plus 1 times x plus 3, giving us 2x plus 1 times x plus 3.
Perfect square trinomials are special quadratics that factor into perfect squares. The pattern is a squared plus 2ab plus b squared equals a plus b squared, or a squared minus 2ab plus b squared equals a minus b squared. For x squared plus 6x plus 9, we see x squared and 9 are perfect squares, and 6x equals 2 times x times 3, so it factors as x plus 3 squared. Similarly, 4x squared minus 12x plus 9 factors as 2x minus 3 squared.
The difference of squares is another special case where we have a squared minus b squared, which factors as a minus b times a plus b. For x squared minus 16, we recognize this as x squared minus 4 squared, so it factors as x minus 4 times x plus 4. Similarly, 9x squared minus 25 becomes 3x minus 5 times 3x plus 5, and 4x squared minus 49y squared factors as 2x minus 7y times 2x plus 7y.
Let's summarize our factoring strategy. First, always look for a greatest common factor. Then count the terms: two terms usually means difference of squares, three terms could be a perfect square or require the standard or AC method. Here are some quick examples: x squared minus 9 factors as x minus 3 times x plus 3. X squared plus 8x plus 16 is a perfect square: x plus 4 squared. For 2x squared plus 5x plus 3, we get 2x plus 3 times x plus 1. And don't forget to factor out common factors first, like 3x squared minus 12 becomes 3 times x minus 2 times x plus 2. Always verify your answer by expanding back to the original expression!