# High School Math : Intermediate Single-Variable Algebra

## Example Questions

### Example Question #47 : Quadratic Equations And Inequalities

Solve the following equation using the quadratic form:

Explanation:

Factor and solve:

or

This has no solutions.

Therefore there is only one solution:

### Example Question #48 : Quadratic Equations And Inequalities

Solve the following equation using the quadratic form:

Explanation:

Factor and solve:

or

Therefore the equation has four solutions:

### Example Question #49 : Quadratic Equations And Inequalities

Solve the following equation using the quadratic form:

Explanation:

Factor and solve:

or

Therefore the equation has two solutions.

### Example Question #50 : Quadratic Equations And Inequalities

Solve the following equation using the quadratic form:

Explanation:

Factor and solve:

Each of these factors gives solutions to the equation:

### Example Question #21 : Solving Quadratic Equations

The product of two consecutive positive numbers is .  What is the sum of the two numbers?

Explanation:

Let the first number and the second number.

The equation to sovle becomes , or .

Factoring we get , so the solution is .  The problem states that the numbers are positive, so the correct numbers are and , which sum to .

### Example Question #52 : Quadratic Equations And Inequalities

Two positive, consecutive odd numbers have a product of .  What is their sum?

Explanation:

Let first odd number and second odd number. Then:

Use the distributive property and subtract from both sides to get .

Factoring we get .

Solving we get , so .

The problem stated that the numbers were positive so the answer becomes .

### Example Question #53 : Quadratic Equations And Inequalities

Find the sum of the solutions to:

Explanation:

Multiply both sides of the equation by , to get

This can be factored into the form

So we must solve

and

to get the solutions.

The solutions are:

and their sum is   .

### Example Question #31 : Solving Quadratic Equations

Find the vertex of the parabola by completing the square.

Explanation:

To find the vertex of a parabola, we must put the equation into the vertex form:

The vertex can then be found with the coordinates (h, k).

To put the parabola's equation into vertex form, you have to complete the square. Completing the square just means adding the same number to both sides of the equation -- which, remember, doesn't change the value of the equation -- in order to create a perfect square.

Put all of the  terms on one side:

Now we know that we have to add something to both sides in order to create a perfect square:

In this case, we need to add 4 on both sides so that the right-hand side of the equation factors neatly.

Now we factor:

Once we isolate , we have the equation in vertex form:

Thus, the parabola's vertex can be found at .

### Example Question #32 : Solving Quadratic Equations

Complete the square:

Explanation:

Begin by dividing the equation by  and subtracting  from each side:

Square the value in front of the  and add to each side:

Factor the left side of the equation:

Take the square root of both sides and simplify:

### Example Question #33 : Solving Quadratic Equations

Use factoring to solve the quadratic equation: