### All PSAT Math Resources

## Example Questions

### Example Question #891 : Algebra

**Possible Answers:**

**Correct answer:**

### Example Question #1 : Distributive Property

If , what is the value of ?

**Possible Answers:**

**Correct answer:**

Remember that (*a – **b *)(*a *+ *b *) = *a *^{2 }– *b *^{2}.

We can therefore rewrite (3*x –* 4)(3*x *+ 4) = 2 as (3*x *)^{2 }– (4)^{2 }= 2.

Simplify to find 9*x*^{2 }– 16 = 2.

Adding 16 to each side gives us 9*x*^{2 }= 18.

### Example Question #2 : Distributive Property

If and , then which of the following is equivalent to ?

**Possible Answers:**

**Correct answer:**

We are asked to find the difference between g(h(x)) and h(g(x)), where g(x) = 2x^{2} – 2 and h(x) = x + 4. Let's find expressions for both.

g(h(x)) = g(x + 4) = 2(x + 4)^{2} – 2

g(h(x)) = 2(x + 4)(x + 4) – 2

In order to find (x+4)(x+4) we can use the FOIL method.

(x + 4)(x + 4) = x^{2} + 4x + 4x + 16

g(h(x)) = 2(x^{2} + 4x + 4x + 16) – 2

g(h(x)) = 2(x^{2} + 8x + 16) – 2

Distribute and simplify.

g(h(x)) = 2x^{2} + 16x + 32 – 2

g(h(x)) = 2x^{2} + 16x + 30

Now, we need to find h(g(x)).

h(g(x)) = h(2x^{2} – 2) = 2x^{2} – 2 + 4

h(g(x)) = 2x^{2} + 2

Finally, we can find g(h(x)) – h(g(x)).

g(h(x)) – h(g(x)) = 2x^{2} + 16x + 30 – (2x^{2} + 2)

= 2x^{2} + 16x + 30 – 2x^{2} – 2

= 16x + 28

The answer is 16x + 28.

### Example Question #2 : Distributive Property

The sum of two numbers is . The product of the same two numbers is . If the two numbers are each increased by one, the new product is . Find in terms of *.*

**Possible Answers:**

**Correct answer:**

Let the two numbers be *x* and *y*.

*x* + *y* = * s*

*xy* = *p*

(*x* + 1)(*y* + 1) = *q*

Expand the last equation:

*xy* + *x* + *y* + 1 = *q*

Note that both of the first two equations can be substituted into this new equation:

*p* + *s* + 1 = *q*

Solve this equation for *q – p* by subtracting *p* from both sides:

*s* + 1 = *q* – *p*

### Example Question #2 : Distributive Property

Expand the expression:

**Possible Answers:**

**Correct answer:**

When using FOIL, multiply the first, outside, inside, then last expressions; then combine like terms.

### Example Question #1 : Distributive Property

Expand the following expression:

**Possible Answers:**

**Correct answer:**

Which becomes

Or, written better

### Example Question #1 : How To Use Foil In The Distributive Property

Which of the following is equal to the expression ?

**Possible Answers:**

**Correct answer:**

Multiply using FOIL:

First = 3x(2x) = 6x^{2}

Outter = 3x(4) = 12x

Inner = -1(2x) = -2x

Last = -1(4) = -4

Combine and simplify:

6x^{2} + 12x - 2x - 4 = 6x^{2} +10x - 4

### Example Question #3 : How To Use Foil In The Distributive Property

Simplify the expression.

**Possible Answers:**

^{}

None of the other answers

**Correct answer:**

Solve by applying FOIL:

First: 2x^{2} * 2y = 4x^{2}y

Outer: 2x^{2} * a = 2ax^{2}

Inner: –3x * 2y = –6xy

Last: –3x * a = –3ax

Add them together: 4x^{2}y + 2ax^{2} – 6xy – 3ax

There are no common terms, so we are done.

### Example Question #3 : Distributive Property

Given the equation above, what is the value of ?

**Possible Answers:**

**Correct answer:**

Use FOIL to expand the left side of the equation.

From this equation, we can solve for , , and .

Plug these values into to solve.

### Example Question #4 : Distributive Property

Expand and simplify the expression.

**Possible Answers:**

**Correct answer:**

We can solve by FOIL, then distribute the . Since all terms are being multiplied, you will get the same answer if you distribute the before using FOIL.

First:

Inside:

Outside:

Last:

Sum all of the terms and simplify. Do not forget the in front of the quadratic!

Finally, distribute the .

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