GRE Quantitative Reasoning › Algebra
Which of the following is a root of the function ?
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x = .
The answer is therefore .
Find one possible value of , given the following equation:
Cannot be determined from the information given.
We begin with the following:
This can be rewritten as
Recall that if you have two exponents with equal bases, you can simply set the exponents equal to eachother. Do so to get the following:
Solve this to get t.
Pets Plus makes bird houses. Their monthly fixed expenses are $750. The cost for each bird house is $15. The bird houses sell for $40.
If Pets Plus sells 50 bird houses, what is the profit?
Let = the number of birdhouses sold each month.
Substituting in 50 for gives an answer of 500, so the profit on 50 birdhouses is $500.
Pets Plus makes bird houses. Their monthly fixed expenses are $750. The cost for each bird house is $15. The bird houses sell for $40.
If Pets Plus sells 50 bird houses, what is the profit?
Let = the number of birdhouses sold each month.
Substituting in 50 for gives an answer of 500, so the profit on 50 birdhouses is $500.
and
are both integers.
If ,
, and
, which of the following is a possible value of
?
Take the values of y that are possible, i.e. 2 and 3, and plug them into the first inequality. First, plug in 2. 2 – 3x > 21. Subtract 2 from both sides, and then divide by –3. Don't forget that when you divide or multiply by a negative number in an inequality you must flip the inequality sign. Thus, x < –19/3. Now plug in 3. We find, following the same steps, that when y=3, x < –6. Thus –7 is the correct answer.
Pets Plus makes bird houses. Their monthly fixed expenses are $750. The cost for each bird house is $15. The bird houses sell for $40.
If Pets Plus sells 50 bird houses, what is the profit?
Let = the number of birdhouses sold each month.
Substituting in 50 for gives an answer of 500, so the profit on 50 birdhouses is $500.
Which of the following is a root of the function ?
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x = .
The answer is therefore .
Three consecutive even numbers add to 42. What is the middle number?
Let = 1st even number,
= 2nd even number, and
= 3rd even number.
Then the equation to solve becomes .
Thus , so the middle number is 14.
Which of the following is a root of the function ?
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x = .
The answer is therefore .