Understanding factoring - GMAT Quantitative

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

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Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by adding both sides:

$\underline{x-2y= 8}$

$\underline{x-2y= -8}$

$\underline{x-2y= -8}$

$\underline{x-2y= 8}$

The set of possible values of is $\{-9, -1, 1, 9 \}$ .

← Didn't Know|Knew It →

Question

$x^{2}+4xy +4 y^{2} = 100$

$x^{2}-4xy +4 y^{2} = 64$

Which of the following is the set of possible values of ?

Tap to reveal answer

Answer

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

$x^{2}+4xy +4 y^{2} = 100$

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

Either or

Similarly,

$x^{2}-4xy +4 y^{2} = 64$

$(x-2y)^{2} = 64$

Either or

Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:

$\underline{x-2y= 8}$

$y = \frac{1}{2}$

$\underline{x-2y= -8}$

$y =- \frac{1}{2}$

$\underline{x-2y= -8}$

$y = \frac{9}{2}$

$\underline{x-2y= 8}$

$y =- \frac{9}{2}$

The set of possible values of is $\left \{ -\frac{9}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{9}{2} \right \}$ .

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Question

Solve for :

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Answer

and

and

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Question

Factor $4y^{2}+4y-15$ .

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Answer

To factor this, we need two numbers that multiply to $\dpi{100} \small 4\times -15=60$ and sum to $\dpi{100} \small 4$ . The numbers $\dpi{100} \small -6$ and $\dpi{100} \small 10$ work.

$4y^{2}+4y-15 = 4y^{2} - 6y + 10y - 15 = 2y(2y-3) + 5(2y-3) = (2y-3)(2y+5)$

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Question

Factor $\frac{y-x}{x^{2}-y^{2}}$ .

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Answer

${x^{2}-y^{2}}$ is a difference of squares. The difference of squares formula is

${a^{2}-b^{2}} = (a-b)(a+b)$

So ${x^{2}-y^{2}} = (x-y)(x+y)$ .

Then, $\frac{y-x}{x^{2}-y^{2}} = \frac{y-x}{(x+y)(x-y)} = \frac{-(x-y)}{(x+y)(x-y)} = \frac{-1}{x+y}$ .

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Question

Solve $x^{2}-6x+5>0$ .

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Answer

Let's factor the expression: $x^{2}-6x+5 = (x-1)(x-5)$ .

We need to look at the behavior of the function to the left and right of 1 and 5. To the left of ,

$x^{2}-6x+5>0$

You can check this by plugging in any value smaller than 1. For example, if ,

,

which is greater than 0.

When takes values in between 1 and 5, $x^{2}-6x+5 <0$ . Again we can check this by plugging in a number between 1 and 5.

, which is less than 0, so no numbers between 1 and 5 satisfy the inequality.

When takes values greater than 5, $x^{2}-6x+5>0$ .

To check, let's try . Then:

so numbers greater than 5 also satisfy the inequality.

Therefore $x>5\ or\ x<1$ .

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