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

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

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Example Question #1 : Algebraic Concepts

Solve for x:

\dpi{100} 3x^{2}+4=31

Possible Answers:

\dpi{100} \pm 9

\dpi{100} 9

\dpi{100} \pm 3

\dpi{100} \pm 1

Correct answer:

\dpi{100} \pm 3

Explanation:

First, subtract 4 from both sides:

\dpi{100} 3x^{2}+4-4=31-4

\dpi{100} 3x^{2}=27

Next, divide both sides by 3:

\dpi{100} \frac{3x^{2}}{3}=\frac{27}{3}

\dpi{100} x^{2}=9

Now take the square root of both sides:

\dpi{100} \sqrt{x^{2}}=\sqrt{9}

\dpi{100} x=\pm 3

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

Solve for \dpi{100} x:

\dpi{100} \frac{x}{3}+22 = 45

Possible Answers:

\dpi{100} 135

\dpi{100} 3

\dpi{100} 66

\dpi{100} 69

Correct answer:

\dpi{100} 69

Explanation:

\dpi{100} \frac{x}{3}+22 = 45

\dpi{100} \frac{x}{3}+22-22 = 45-22

\dpi{100} \frac{x}{3} = 23

\dpi{100} (3)\cdot \frac{x}{3} = 23\cdot (3)

\dpi{100} x=69

Example Question #3 : Algebraic Concepts

Solve for :

Possible Answers:

Correct answer:

Explanation:

Rewrite this as a compound statement and solve each separately:

 

 

Example Question #4 : Algebraic Concepts

Solve for 

Possible Answers:

Correct answer:

Explanation:

FOIL each of the two expressions, then solve:

Solve the resulting linear equation:

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

Solve for :

Possible Answers:

Correct answer:

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 #6 : Algebraic Concepts

Solve for :

Possible Answers:

Correct answer:

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 #7 : Equations

Solve for :

Give all solutions.

Possible Answers:

Correct answer:

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 #7 : Algebraic Concepts

Solve for :

Possible Answers:

The equation has no solution.

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

Correct answer:

Explanation:

Simplify both sides, then solve:

Example Question #9 : Equations

Possible Answers:

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

The equation has no solution.

Correct answer:

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.

Example Question #10 : Equations

Solve for :

Possible Answers:

Correct answer:

Explanation:

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