How to find the solution to a quadratic equation

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.

Since , .

Thus

Of these two, only 4 is a possible answer.

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) = 2_x_2 – 4_x_ + 1

f(k) = 2_k_2 – 4_k_ + 1 = 31

Subtract 31 from both sides.

2_k_2 – 4_k –_ 30 = 0

Divide both sides by 2.

k_2 – 2_k – 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.

k_2 –2_k – 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) = (_x_2 + 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.

Factor the expression and set each factor equal to 0:

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 .

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 .

Let = first positive number and = second positive number

So the equation to solve becomes

Using the distributive property, multiply out the equation and then set it equal to 0. Next factor to solve the quadratic.

Begin by distributing the three on the right side of the equation:

Next combine your like terms by subtracting from both sides to give you

Next, subtract 9 from both sides to give you . To solve for , now take the square root of both sides. This gives you the answer,

The corresponding sides of similar triangles are in proportion, so we can set up and solve the proportion statement for :

, so

For the sake of simplicuty, we will let

Since is the midpoint of , .

Also, .

The proportion statement becomes

Solve for using cross-products:

By the quadratic equation, setting :

There are two possibilities:

or

is divided into segments of length 2.9 and 17.1. The lesser is the length of , so the correct choice is 2.9.

Let = the first positive number and = the second positive number.

So the equation to solve is

We multiply out the equation and set it equal to zero before factoring.

thus the two numbers are 21 and 24 for a sum of 45.