To factor a quadratic where the second-order term has a coefficient of 1, find two numbers that multiply up to the constant, –7, and add up to the first-order term’s coefficient, 6. These two numbers are 7 and –1, so you can rewrite the quadratic equation as follows:
To find the solutions—or roots of the resulting quadratic function—set the equation equal to zero. You can see that this happens when x = –7 or x = 1. (F) is the only answer choice that lists one of these options.
If you chose (H) or (J), you probably forgot that a root is the negative of one of those two numbers you found, because it’s a value of x that makes one of the bracketed expressions equal to zero.
If you got any of the other answer choices, you found a term from the right sequence, but not the 21st term.
Taking the second inequality, x > 1, into account, you know that the number line needs to show values between 1 and 4, including 4 and excluding 1. The only answer choice representing this is (E).
You can eliminate (G) because 2x × 2x does not equal 2x2, a.
You can eliminate (J) and (K) because neither (–1) × (–1) nor 1 × 1 equals –1, c.
You can eliminate (F) because (1 × (–1)) + (1 × 2) = –1 + 2 = 1, and b should equal –1.
You can substitute this back into the equations to confirm your answer.
Note that this is a bit tricky, because you have to see that I is possible even while the volume in the tank is increasing. The volume is not increasing as much as before, so I may be the result of this.
This means that after 5 hours, (8 boards per hour)(5 hours) = 40 boards.
Next, find the number of boards that would have been produced with the old method:
The difference between the two methods is 40 – 10 = 30 boards.
If you got any other result, check your algebra as it is most likely due to incorrect simplifying and elimination.
On the second day, Lucy collects 4 cans; on the third, 8; on the fourth, 16; on the fifth, 32. Adding all the cans together gives 2 + 4 + 8 + 16 + 32 = 62 cans in total.
If you chose (H), you probably picked the number of cans collected on the fifth day alone, rather than the cumulative number of cans collected.
Substituting V into the first equation:
If you chose another answer, make sure to review your log rules.