GMAT Math : Algebra

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Example Questions

Example Question #43 : Simplifying Algebraic Expressions

Simplify:

$(x+10)(x-10)(x^{2}+100)- (x+5)(x-5)(x^{2}+25)$

Possible Answers:

- 9,375

-5,000

9,375

875

Correct answer:

- 9,375

Explanation:

The difference of squares pattern can be applied twice to each product.

$(x+5)(x-5)(x^{2}+25)$

$= (x^{2}-5^{2})(x^{2}+25)$

$= (x^{2}-25)(x^{2}+25)$

$=\left (x^{2} \right )^{2}-25^{2}$

$= x^{4}-625$

$(x+10)(x-10)(x^{2}+100)$

$= (x^{2}-10^{2})(x^{2}+100)$

$= (x^{2}-100)(x^{2}+100)$

$=\left (x^{2} \right )^{2}-100^{2}$

$= x^{4}-10,000$

$(x+10)(x-10)(x^{2}+100)- (x+5)(x-5)(x^{2}+25)$

$= \left (x^{4}-10,000 \right ) - (x^{4}-625)$

$= x^{4}- x^{4} -10,000 +625$

= - 9,375

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Example Question #41 : Simplifying Algebraic Expressions

Simplify.

4x^2+3xy-(x-6y+2)^2

Possible Answers:

3x^2-4x+15xy-24y-36y^2-4

3x^2-4x+15xy+12y-36y^2-4

3x^2-4x+15xy+24y-36y^2-4

5x^2-4x+15xy-24y-36y^2+4

Correct answer:

3x^2-4x+15xy+24y-36y^2-4

Explanation:

4x^2+3xy-(x-6y+2)^2

4x^2+3xy-(x^2-6xy+2x-6xy+36y^2-12y+2x-12y+4)

Simplify what is in the parentheses:

4x^2+3xy-(x^2-12xy+4x+36y^2-24y+4)

Distribute the negative sign outside:

4x^2+3xy-x^2+12xy-4x-36y^2+24y-4

Lastly, combine like terms:

3x^2-4x+15xy+24y-36y^2-4

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Example Question #42 : Simplifying Algebraic Expressions

Simplify:

$\frac{3x^3 - 60x^2 - 900x}{3x^2 - 60x}$

Possible Answers:

$\frac{3x}{x-30}$

$\frac{1}{x-30}$

$\frac{x^2 - 20x + 300}{x-30}$

x + 10

Correct answer:

x + 10

Explanation:

The first thing we can do is factor out 3x from both the top and bottom of the expression:

$\frac{3x^3 - 60x^2 - 900x}{3x^2 - 60x} = \frac{3x(x^2 - 20x -300)}{3x(x-30)}$

We can then factor the numerator's polynomial:

$\frac{3x(x-30)(x+10)}{3x(x-30)}$

3x divided by itself and (x-30) divided by itself both cancel to 1, leaving x + 10 as the answer.

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Example Question #43 : Simplifying Algebraic Expressions

Simplify the following.

-(2x+4y^2)^2+(3x^2-2y+7y^2+y^4)

Possible Answers:

$7x^{2}+32xy^{2}+17y^{8}-2y+7$

-x^2-16xy^2-15y^4-2y+7y^2

7x^2+8xy^2+17y^4-2y+7y^2

7x^2+16xy^2+17y^4-2y+7y^2

-x^2-8xy^2-15y^4-2y+7y^2

Correct answer:

-x^2-16xy^2-15y^4-2y+7y^2

Explanation:

We can begin by expanding the first parentheses:

-(2x+4y^2)^2+(3x^2-2y+7y^2+y^4)

-(4x^2+8xy^2+8xy^2+16y^4)+(3x^2-2y+7y^2+y^4)

We can now combine the like terms: -(4x^2+16xy^2+16y^4)+(3x^2-2y+7y^2+y^4)

Distribute the negative sign:

-4x^2-16xy^2-16y^4+3x^2-2y+7y^2+y^4

Lastly, combine like terms:

-x^2-16xy^2-15y^4-2y+7y^2

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Example Question #271 : Algebra

The first two terms of an arithmetic sequence are, in order, and $W^{2}$ . What is the third term?

(Assume is positive.)

Possible Answers:

$W^{2} -W$

$W^{4}$

$W^{2} +W$

$W^{3}$

$2W^{2} -W$

Correct answer:

$2W^{2} -W$

Explanation:

The common difference of an arithmetic sequence such as this is the difference of the second and first terms:

$d = W^{2}- W$

Add this to the second term to obtain the third term:

$W^{2}+ W^{2}- W = 2W^{2}- W$

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Example Question #1356 : Problem Solving Questions

The first two terms of an arithmetic sequence are, in order, $\left (W + 3 \right )^{2}$ and $\left (W + 5 \right )^{2}$ . Which of the following is the third term of the sequence?

Possible Answers:

$W^{2}+ 14W + 49$

$W^{2}+ 14W +36$

$2 W^{2}+ 16W + 34$

$W^{2}+ 14W + 41$

$2 W^{2}+ 16W + 41$

Correct answer:

$W^{2}+ 14W + 41$

Explanation:

The terms can be rewritten by squaring each binomial as follows:

The first term is

$\left (W + 3 \right )^{2} =W^{2} + 2 \cdot W \cdot 3 + 3 ^{2} = W^{2} + 6W + 9$

The second term is

$\left (W + 5 \right )^{2} =W^{2} + 2 \cdot W \cdot 5 + 5 ^{2} = W^{2} + 10W + 25$

The common difference of an arithmetic sequence such as this is the difference of the second and first terms:

$\begin{matrix} W^{2} + 10W + 25\\ \underline{W^{2} + \; 6W + \; \; 9} \\\; \; \; \; \; \; \; \; \; \; \; \; 4W+16\end{matrix}$

Add this to the second term to obtain the third:

$\begin{matrix} W^{2} + 10W + 25\\ \underline{ \; \; \; \; \; \; \; \; \; \; \; \; 4W+16} \\W^{2}+ 14W + 41 \end{matrix}$

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Example Question #271 : Algebra

If $x=\sqrt{\frac{25}{100}}$ ,

what is the value of

Possible Answers:

$\frac{1}{5}$

$\frac{5}{10}$

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{2}{5}$

Correct answer:

$\frac{1}{2}$

Explanation:

Simplify.

$x=\sqrt{\frac{25}{100}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

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Example Question #42 : Simplifying Algebraic Expressions

If you were to write $(x-3)^{10}$ in expanded form in descending order of degree, what would the third term be?

Possible Answers:

$810x^{8}$

$405x^{8}$

$45x^{8}$

$243x^{8}$

$945x^{8}$

Correct answer:

$405x^{8}$

Explanation:

By the Binomial Theorem, if you expand $(A+B)^{n}$ , writing the result in standard form, the $r \mathrm{th}$ term (with the terms being numbered from 0 to ) is

$C(n,r)A^{n-r}B^{r}$

Set A=x , B=-3 , n=10 , and r=2 (again, the terms are numbered 0 through , so the third term is numbered 2) to get

$C(10,2)\cdot x^{10-2}\cdot (-3)^{2}$

$=\frac{10!}{8! 2!}\cdot x^{8}\cdot 9$

$=45\cdot x^{8}\cdot 9$

$=405x^{8}$

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Example Question #272 : Algebra

Assume that $x,y \neq 0$ .

Which of the following expressions is equal to the following expression?

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

Possible Answers:

$4x^{2}y^{2}$

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

$\frac{2 }{x} + \frac{2 }{y}$

$\frac{2 }{x^{2}y} + \frac{2 }{xy^{2}}$

Correct answer:

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

Explanation:

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

$=\frac{(x^{2}+2xy+y^{2}) + (x^{2}-2xy+y^{2}) }{x^{2}y^{2}}$

$=\frac{x^{2}+x^{2}+2xy-2xy+y^{2}+y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2}+2y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2} }{x^{2}y^{2}} + \frac{2y^{2} }{x^{2}y^{2}}$

$=\frac{2 }{y^{2}} + \frac{2 }{x^{2}} = \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

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Example Question #1 : Linear Equations, One Unknown

For what value of $\small N$ would the following equation have no solution?

$\small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

Possible Answers:

$\small \small N= -1$

$\small N= -2$

$\small \small N= 1$

$\small N= 2$

The equation must always have at least one solution regardless of the value of $\small N$ .

Correct answer:

$\small N= 2$

Explanation:

Simplify both sides of the equation as much as possible, and solve for $\small x$ in the equation in terms of $\small N$ :

$\small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

$\small 3 \cdot 4x - 3 \cdot 7 + 12 = 2 \cdot 5x - 2 \cdot 3 + N \cdot x - N \cdot 3$

$\small 12x - 21 + 12 = 10x - 6 + Nx - 3N$

$\small 12x - 9 = (10+N)x + (- 6 - 3N)$

$\small \small 12x - (10+N)x = (- 6 - 3N) + 9$

$\small (2-N)x = 3 - 3N$

$\small x = \frac{3 - 3N}{2 -N}$

$\small x$ has exactly one solution unless the denominator is 0 - that is, $\small N = 2$ . We make sure that this value renders no solution by substituting:

$\small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

$\small \small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + 2 (x - 3)$

$\small 12x - 21 + 12 = 10x - 6 + 2x - 6$

$\small 12x - 9 = 12x - 12$

$\small - 9 = - 12$

The equation has no solution, and $\small N = 2$ is the correct answer.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #43 : Simplifying Algebraic Expressions

Example Question #41 : Simplifying Algebraic Expressions

Example Question #42 : Simplifying Algebraic Expressions

Example Question #43 : Simplifying Algebraic Expressions

Example Question #271 : Algebra

Example Question #1356 : Problem Solving Questions

Example Question #271 : Algebra

Example Question #42 : Simplifying Algebraic Expressions

Example Question #272 : Algebra

Example Question #1 : Linear Equations, One Unknown

All GMAT Math Resources

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