Define the variables to be x = first multiple of three and x + 3 = the next consecutive multiple of 3.

Knowing the product of these two numbers is 54 we get the equation x(x + 3) = 54. To solve this quadratic equation we need to multiply it out and set it to zero then factor it. So x² + 3x – 54 = 0 becomes (x + 9)(x – 6) = 0.  Solving for x we get x = –9 or x = 6 and only the positive number is correct.  So the two numbers are 6 and 9 and their sum is 15.

Generally, quadratic equations have two answers.

First, the equations must be put in standard form: 3x² + 10x + 3 = 0

Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.

Third, check the answer by plugging the answers back into the original equation.

Begin by getting our equation into the form Ax² + BX + C = 0:

3x² – 11x + 10 = 0

Now, if you factor the left, you can find the answer. Begin by considering the two groups.  They will have to begin respectively with 3 and 1 as coefficients for your x value.  Likewise, looking at the last element, you can tell that both will have to have a + or –, since the C coefficient is positive.  Finally, since the B coefficient is negative, we know that it will have to be –. We know therefore:

(3x – ?)(x – ?)

The potential factors of 10 are: 10, 1; 1, 10; 2, 5; 5, 2

5 and 2 work:

(3x – 5)(x – 2) = 0 because you can FOIL (3x – 5)(x – 2) back into 3x² – 11x + 10.

Now, the trick remaining is to set each of the factors equal to 0 because if either group is 0, the whole equation will be 0:

3x – 5 = 0 → 3x = 5 → x = 5/3

x – 2 = 0 → x = 2

Therefore, x is either 5 / 3 or 2. The former is presented as an answer.

The equation we are asked to solve is 16^x – 10(4)^x + 16 = 0.

Equations of this type can often be "transformed" into other equations, such as linear or quadratic equations, if we rewrite some of the terms. 

First, we can notice that 16 = 4². Thus, we can write 16^x as (4²)^x or as (4^x)².

Now, the equation is (4^x)² – 10(4)^x + 16 = 0

Let's introduce the variable u, and set it equal 4^x. The advantage of this is that it allows us to "transform" the original equation into a quadratic equation.

u² – 10u + 16 = 0

This is an equation with which we are much more familiar. In order to solve it, we need to factor it and set each factor equal to zero. In order to factor it, we must think of two numbers that multiply to give us 16 and add to give us –10. These two numbers are –8 and –2. Thus, we can factor u² – 10u + 16 = 0 as follows:

(u – 8)(u – 2) = 0

Next, we set each factor equal to 0.

u – 8 = 0

Add 8.

u = 8

u – 2 = 0

Add 2.

u = 2.

Thus, u must equal 2 or 8. However, we want to find x, not u. Since we defined u as equal to 4^x, the equations become:

4^x = 2 or 4^x = 8

Let's solve 4^x = 2 first. We can rewrite 4^x as (2²)^x = 2^2x, so that the bases are the same.

2^2x = 2 = 2¹

2x = 1

x = 1/2

Finally, we will solve 4^x = 8. Once again, let's write 4^x as 2^2x. We can also write 8 as 2³.

2^2x = 2³

2x = 3

x = 3/2

The original question asks us to find the sum of the values of x that solve the equation. Because x can be 1/2 or 3/2, the sum of 1/2 and 3/2 is 2.

The answer is 2. 

There are two values where x² = x, namely x = 0 and x = 1. All values between 0 and 1 get smaller after squaring. All values greater than 1 get larger after squaring.

In order to find the value of f(g(k)), we will first need to find k. We are told that f(k) = 31, so we can write an expression for f(k) and solve for k.

f(x) = 2x² – 4x + 1

f(k) = 2k² – 4k + 1 = 31

Subtract 31 from both sides.

2k² – 4k – 30 = 0

Divide both sides by 2.

k² – 2k – 15 = 0

Now, we can factor this by thinking of two numbers that multiply to give –15 and add to give –2. These two numbers are –5 and 3.

k² –2k – 15 = (k – 5)(k + 3) = 0

We can set each factor equal to 0 to find the values for k.

k – 5 = 0

Add 5 to both sides.

k = 5

Now we set k + 3 = 0.

Subtract 3 from both sides.

k = –3

This means that k could be either 5 or –3. However, we are told that k is a negative number, which means k = –3.

Finally, we can evaluate the expression f(g(–3)). First we need to find g(–3).

g(x) = (x² + 16)^(1/2)

g(–3) = ((–3)² + 16)^(1/2)

= (9 + 16)^(1/2)

= 25^(1/2)

Raising something to the one-half power is the same as taking the square root.

25^(1/2) = 5

Now that we know g(–3) = 5, we must find f(5).

f(5) = 2(5)² – 4(5) + 1

= 2(25) – 20 + 1 = 31

The answer is 31.

The equation in the problem is quadratic, so we can use the quadratic formula to solve it. If an equation is in the form ax² + bx + c = 0, where a, b, and c are constants, then the quadratic formula, given below, gives us the solutions of x.

In this particular problem, a = 2, b = –6, and c = 3.

The value under the square-root, b² – 4ac, is called the discriminant, and it gives us important information about the nature of the solutions of a quadratic equation.

If the discriminant is less than zero, then the roots are not real, because we would be forced to take the square root of a negative number, which yields an imaginary result. The discriminant of the equation we are given is (–6)² – 4(2)(3) = 36 – 24 = 12 > 0. Because the discriminant is not negative, the solutions to the equation will be real. Thus, option I is correct.

The discriminant can also tell us whether the solutions of an equation are rational or not. If we take the square root of the discriminant and get a rational number, then the solutions of the equation must be rational. In this problem, we would need to take the square root of 12. However, 12 is not a perfect square, so taking its square root would produce an irrational number. Therefore, the solutions to the equation in the problem cannot be rational. This means that choice II is incorrect.

Lastly, the discriminant tells us if the roots to an equation are distinct (different from one another). If the discriminant is equal to zero, then the solutions of x become (–b + 0)/2a and (–b – 0)/2a, because the square root of zero is 0. Notice that (–b + 0)/2a is the same as (–b – 0)/2a. Thus, if the discriminant is zero, then the roots of the equation are the same, i.e. indistinct. In this particular problem, the discriminant = 12, which doesn't equal zero. This means that the two roots will be different, i.e. distinct. Therefore, choice III applies.

The answer is choices I and III only.

First let's see if there is a common term. 

3x² + 15x – 18 = 0

We can pull out a 3: 3(x² + 5x – 6) = 0

Divide both sides by 3: x² + 5x – 6 = 0

We need two numbers that sum to 5 and multiply to –6. 6 and –1 work.

(x + 6)(x – 1) = 0

x = –6 or x = 1

Factor the expression and set each factor equal to 0:

Let  = the first number and  = the second number.

So the equation to solve becomes .  This quadratic equation needs to be multiplied out and set equal to zero before factoring.  Then set each factor equal to zero and solve.  Only positive numbers are correct, so the answer is .