The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x² = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

Tommy's current age is represented by t, and Sara's is represented by s. In five years, both Tommy's and Sara's ages will be increased by five. Thus, in five years, we can represent Tommy's age as and Sara's as .

The problem tells us that Tommy's age in five years will be twice as great as Sara's in five years. Thus, we can write an algebraic expression to represent the problem as follows:

In order to solve for t, first simplify the right side by distributing the 2.

Then subtract 5 from both sides.

The answer is .

First we need the area or the rectangle. 24 * 8 = 192. So now we know that 10 gallons will cover 75 ft² and x gallons will cover 192 ft². We set up a simple ratio and cross multiply to find that 75x = 1920.

x = 25.6

The best way to solve this problem is to translate it into an equation, "decreased" meaning subtract and "increased" meaning add:

x – 7 = 10 + 7

x = 24

First find 2%3 = (2 * 3 + 3 * 2)/(6 * 2 * 3) = 12/36 = 1/3, then 3%2 = (2 * 2 + 3 * 3)/(6 * 3 * 2) = 13/36 which is greater.

5 more than 5 = 10

5 + x = 10

Subtract 5 from each side of the equation: x = 5 → 2x = 10

First, we need to find the possible values of k such that f(k) = 0.

We can use the definition of f(x) to write an expression for k.

f(x) = 2x³ + 7x² – 4x

f(k) = 2k³ + 7k² – 4k = 0

In order to solve this equation, we will want to factor as much as we can. We can immediately see that we could take out a k from all three terms.

2k³ + 7k² – 4k = 0

k(2k² + 7k – 4) = 0

Now, we need to factor 2k² + 7k – 4 by grouping. When we multiply the outer two coefficients we get (2)(–4) = –8. The middle coefficient is 7. This means we need to think of two numbers that multiply to give us –8, but add to give us 7. These two numbers are 8 and –1. We can now rewrite 2k² + 7k – 4 as follows:

2k² + 7k – 4 = 2k² + 8k – k – 4

Group the first two terms and the second two terms.

2k² + 8k – k – 4 = (2k² + 8k ) + (–k – 4)

Next, factor out a 2k from the first two terms and a –1 from the last two terms.

(2k² + 8k ) + (–k – 4) = 2k(k + 4) + –1(k + 4)

Then, factor out k + 4 from the 2k and the –1.

2k(k + 4) + –1(k + 4) = (k + 4)(2k – 1)

Thus, 2k² + 7k – 4, when fully factored, equals (k + 4)(2k – 1).

Now, let's go back to the equation k(2k² + 7k – 4) = 0 and substitute (k + 4)(2k – 1) for 2k² + 7k – 4.

k(2k² + 7k – 4) = k(k + 4)(2k – 1) = 0

We now have three factors, and we can set each equal to zero to find the possible values of k.

The first factor is k. This means k = 0 is one value for k.

The next factor is k + 4.

k + 4 = 0

k = –4

The last factor is 2k – 1.

2k – 1 = 0

2k = 1

k = 1/2

The values of k for which f(k) = 0 are 0, –4, and 1/2. However, we are told that k is positive, so this means that k can only be 1/2.

Ultimately, the problem asks us to find g(k). This means we must use the equation for g(x) to find g(1/2).

g(x) = 4x – x³

g(1/2) = 4(1/2) – (1/2)³

= 2 – (1/8)

= 15/8

The answer is 15/8.

We need to translate the information given into an algebraic equation that will help us solve for the integers.

Let x represent the smaller of the two integers. Because they are consecutive even integers, the other integer is equal to x + 2.

We are told that the sum of the two integers is two greater than one-tenth of their product. We can represent the sum of the two integers as x + (x + 2).

To represent one-tenth of the product, we can write (1/10)(x)(x + 2). Two greater than this quantity would be equal to 2 + (1/10)(x)(x + 2). Remember that "two greater" requires us to add 2, while "two times greater" would have required us to multiply by 2. 

Now, we can set the sum equal to two greater than one-tenth the product.

x + (x + 2) = 2 + (1/10)(x)(x + 2)

Multiply both sides by 10 to eliminate the fraction.

10(x + x + 2) = 10(2 + (1/10)x(x + 2))

Distribute the 10 on both sides.

10x + 10x + 20 = 20 + 10(1/10)x(x + 2)

Simplify both sides.

20x + 20 = 20 + x(x + 2)

Distribute the x on the right side.

20x + 20 = 20 + x² + 2x

Subtract 20 from both sides.

20x = x² + 2x

Subtract 20x from both sides.

x² – 18x = 0

Factor out an x.

x(x – 18) = 0

Set each factor equal to zero.

x = 0, or x – 18 = 0, which means x = 0 or 18. However, because we are told that the integers are positive, the value of x must be 18.

The smaller integer is 18. We are asked to find the sum of its factors. To find the factors of 18, we must think of all the pairs of numbers that will multiply to give us 18. The pairs are as follows:

1 and 18, 2 and 9, 3 and 6

This means the factors of 18 are 1, 2, 3, 6, 9, and 18. The sum of these numbers is 1 + 2 + 3 + 6 + 9 + 18 = 39.

The answer is 39.