# ISEE Upper Level Math : How to find the solution to an equation

## Example Questions

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### Example Question #1 : How To Find The Solution To An Equation

Solve for x:

Explanation:

First, subtract 4 from both sides:

Next, divide both sides by 3:

Now take the square root of both sides:

Solve for :

Explanation:

### Example Question #2 : Algebraic Concepts

Solve for :

Explanation:

Rewrite this as a compound statement and solve each separately:

### Example Question #3 : How To Find The Solution To An Equation

Solve for

Explanation:

FOIL each of the two expressions, then solve:

Solve the resulting linear equation:

### Example Question #4 : How To Find The Solution To An Equation

Solve for :

Explanation:

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

Now factor the quadratic expression  into two binomial factors , replacing the question marks with two integers whose product is  and whose sum is . These numbers are , so:

or

The solution set is .

### Example Question #3 : Algebraic Concepts

Solve for :

Explanation:

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:

Use the  method to factor the quadratic expression ; we are looking to split the linear term by finding two integers whose sum is  and whose product is . These integers are , so:

Set each expression equal to 0 and solve:

or

The solution set is .

### Example Question #5 : How To Find The Solution To An Equation

Solve for :

Give all solutions.

Explanation:

Rewrite this quadratic equation in standard form:

Factor the expression on the left. We want two integers whose sum is  and whose product is . These numbers are , so the equation becomes

.

Set each factor equal to 0 separately, then solve:

### Example Question #6 : How To Find The Solution To An Equation

Solve for :

The equation has the set of all real numbers as its solution set.

The equation has no solution.

Explanation:

Simplify both sides, then solve:

### Example Question #2 : How To Find The Solution To An Equation

The equation has the set of all real numbers as its solution set.

The equation has no solution.

The equation has the set of all real numbers as its solution set.

Explanation:

Simplify both sides, then solve:

This is an identically true statement, so the original equation has the set of all real numbers as its solution set.

Solve for :