# Algebra II : Finding Roots

## Example Questions

### Example Question #31 : Finding Roots

Solve for x.

Explanation:

This is a factoring problem, so we need to get all of the variables on one side and set the equation equal to zero. To do this we subtract 128 from both sides to get .

We then notice that all four numbers are divisible by four, so we can simplify the expression to .

Think of the equation in this format to help with the following explanation.

We must then factor to find the solutions for x. To do this we must make a factor tree of c (which is 32 in this case) to find the possible solutions. The possible numbers are 1 * 32, 2 * 16, and 4 * 8.

Since c is negative, we know that our factoring will produce a positive and negative number.

We then look at b to see if the greater number will be positive or negative. Since b is positive, we know that the greater number from our factoring tree will be positive.

We then use addition and subtraction with the factoring tree to find the numbers that add together to equal b. Remember that the greater number is positive and the lesser number is negative in this example.

Positive 8 and negative 4 equal b. We then plug our numbers into the factored form of .

We know that anything multiplied by 0 is equal to 0, so we plug in the numbers for x which make each equation equal to 0. In this case .

### Example Question #32 : Finding Roots

Solve for x.

x = –12, –1

x = 4, 1

x = –2/3, –3

x = –4, –1

x = –9, –2

x = –2/3, –3

Explanation:

1) Combine like terms and simplify.

No further simplification is possible. The first term has a coefficient that can't be factored away. FOIL requires that all terms be multiplied by each other at some point, so the presence of the coefficient has to be reflected in every step of the factoring.

2) Practically speaking, that means we add an extra step. Multiply the coefficient of the first term by the last term before factoring.

3 * 6 = 18

Factors of 18 include:

1 + 18 = 19

2 + 9 = 11

3) Now pull out the common factor in each of the pairs, "3x" from the first two and "2" from the second two.

4) Pull out the "(x+3)" from both terms.

5) Set both parts equal to zero and solve.

3x + 2 = 0, x = –2/3

x + 3 = 0, x = –3

### Example Question #31 : Finding Roots

Find the roots of .

Explanation:

First, you need to factor the expression. This takes a little bit of trial and error, but even though the leading coefficient is not 1, there's only one way to get 2 (2 x 1) so that is helpful to remember!

Factored, the expression is .

Once you have those expressions, you can set them equal to 0 to get the roots, or solutions to the equation.

When you set .

And when you set

.

### Example Question #32 : Finding Roots

Solve

Explanation:

You need to first factor the equation. Then you willbe able to set those expressions equal to 0 and find your roots. There is only one way to get your leading coefficient of 3 (3 x 1) so that's helpful when figuring it out.

In the form  remember you want a factor of  that when mulitplied by the factor of  and added together will give you the value of .

Factored, it comes out to:

.

When you set those expressions equal to 0, you get

.

### Example Question #33 : Finding Roots

Find the roots of the given quadratic.

Explanation:

The roots of a quadratic can be found by factoring. Factoring the solution will give you an expression that when multiplying will result in the initial quadratic.

The roots for this expression are . Similarly the roots of a quadratic can be found using the quadratic quadratic formula where the parent function of a quadratic is represented as  and the quadratic formula is .

### Example Question #34 : Finding Roots

What are the roots of the following graph?

None of the above

Explanation:

Roots when looking at a graph are simply the x-values of where the function crosses the x-axis.

Which, when looking at this graph, it is clearly

and

### Example Question #35 : Finding Roots

Solve the equation.

Explanation:

To solve this equation perform the oppisite operation to isolate the variable.

Recall that the square root of a negative number results in an imaginary number.

Therefore,

.

### Example Question #36 : Finding Roots

Solve the equation.

Explanation:

To solve this equation perform the oppisite operation to isolate the variable.

Recall that the square root of a negative number is an imaginary number.

Therefore,

.

### Example Question #37 : Finding Roots

What are the x-intercepts for

Explanation:

The x-intercepts of a quadratic equation are also the solutions. To find them, factor the quadratic equation. After some trial and error, it can be factored to: . Set those expressions equal to  to get you x-intercepts. Your answers are: .

### Example Question #38 : Finding Roots

Find all solutions to the following quadratic equation:

Explanation:

Find all solutions to the following quadratic equation:

We can solve the following equation by first bringing the -225 to the other side:

Next, take the square root of both sides.