Absolute Value Quadratics + Mixture Problems + The Mixture Trick

If you find an Absolute Value Quadratic, the same principle applies as in a linear equation: you set up two equations (one positive and one negative)

Ex: 15 = | x² + 7x |

15 = x² + 7x and -15 = x² + 7x → x² + 7x - 15 =0 and x² +7x + 15 = 0

Mixture Problems: Ex: Solution A contains 30% alcohol and Solution B contains 55% alcohol. If 300 mL of Solution A is mixed with 200 mL of Solution B, then alcohol accounts for what percentage of the combined mixture?

Step 1: Calculate the total amount of the mixture. This will be the denominator of the fraction we use to calculate the percentage

Step 2: Calculate the amount of alcohol in each of the two solutions

Step 3: Add the two numbers from step 2 and put it over the total (from step 1) to get the final answer

The Mixture Trick: Ex: Bill mixes Fab Soil (30% nitrogen by volume) with BackYard Dirt (42% by volume) and finds that the combined soil contains 38% nitrogen by volume. In what ratio were the two soils mixed?

Step 1: Put the lower percentage on the left, the combined percentage in the middle, and the higher percentage on the right

Step 2: Calculate the “gap” or “distance” between each number and the middle number

Step 3: The two “distance” numbers we found in step 2 represent the initial mixing ratio. So that’s 8:4, or 2:1. But which one is 2 and which one is 1? Well, notice how the 38 number is closer to the 42 number. That means that there must be more of the 42% mixture, so we actually have to “flip” the ratio for it to make sense

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