For a rectangle,  and  where  = width and  = length.

So we get two equations with two unknowns:

Making a substitution we get

Solving the quadratic equation we get  or .

The difference is .

Since , .

Thus

Of these two, only 4 is a possible answer.

To solve by factoring, we need two numbers that add to and multiply to .

In order for the equation to equal zero, one of the terms must be equal to zero.

OR

Our final answer is that .

The first thing to notice is that you have powers with a regular sequence.  This means you can simply treat it like a quadratic equation.  You are then able to factor it as follows:

The factoring can quickly be done by noticing that the 14 must be either or .  Because it is negative, one constant will be negative and the other positive.  Finally, since the difference between 14 and 1 cannot be 5, it must be 7 and 2.

Alternatively, one could use the quadratic formula.

The end result is that you have:

The latter of these two gives only complex answers, so there are two real answers.

We can use the quadratic formula to find the solutions to quadratic equations in the form ax² + bx + c = 0. The quadratic formula is given below.

In order for the formula to give us real solutions, the value under the square root, b² – 4ac, must be greater than or equal to zero. Otherwise, the formula will require us to find the square root of a negative number, which gives an imaginary (non-real) result. 

In other words, we need to look at each equation and determine the value of b² – 4ac. If the value of b² – 4ac is negative, then this equation will not have real solutions.

Let's look at the equation 2x² – 4x + 5 = 0 and determine the value of b² – 4ac.

b² – 4ac = (–4)² – 4(2)(5) = 16 – 40 = –24 < 0

Because the value of b² – 4ac is less than zero, this equation will not have real solutions. Our answer will be 2x² – 4x + 5 = 0.

If we inspect all of the other answer choices, we will find positive values for b² – 4ac, and thus these other equations will have real solutions.

We are told that f(x) = x² - 4x + 2, and g(x) = 6 - x. Let's find expressions for f(k) and g(k).

f(k) = k² – 4k + 2

g(k) = 6 – k

Now, we can set these expressions equal.

f(k) = g(k)

k² – 4k +2 = 6 – k

Add k to both sides.

k² – 3k + 2 = 6

Then subtract 6 from both sides.

k² – 3k – 4 = 0

Factor the quadratic equation. We must think of two numbers that multiply to give us -4 and that add to give us -3. These two numbers are –4 and 1.

(k – 4)(k + 1) = 0

Now we set each factor equal to 0 and solve for k.

k – 4 = 0

k = 4

k + 1 = 0

k = –1

The two possible values of k are -1 and 4. The question asks us to find their sum, which is 3.

The answer is 3. 

In order to solve for , we must apply the quadratic formula. The quadratic formula can be defined as  and can be applied to any equation in the form . The equation we are given is indeed in the form , where . 

Substituting into the quadratic formula, we find 

Simplifying, we find

Therefore, 

To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)

(x + 2)(x + 3) = 0

x = –2 or x = –3

Given a quadratic equation equal to zero you can factor the equation and set each factor equal to zero. To factor you have to find two numbers that multiply to make 9 and add to make 6. The number is 3. So the factored form of the problem is (x+3)(x+3)=0. This statement is true only when x+3=0. Solving for x gives x=-3. Since this problem is multiple choice, you could also plug the given answers into the equation to see which one works. 

64x² + 24x - 10 = 0

Factor the equation:

(8x + 5)(8x – 2) = 0

Set each side equal to zero

(8x + 5) = 0

x = -5/8

(8x – 2) = 0

x = 2/8 = 1/4