How to simplify an expression - SAT Math

Multiply each term of the first factor by each term of the second factor and then combine like terms.

(2x + 4)(x2 – 2x + 4) = 2x3 – 4x2 + 8x + 4x2 – 8x + 16 = 2x3 + 16

Add the two equations:

a + b = 10

b + c = 15

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a + b + b + c = 10 + 15

a + 2_b_ + c = 25 (this is the denominator of the answer)

Subtract the two equations:

b + c = 15

a + b = 10

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b + c – (a + b) = 15 – 10

c – a = 5 (this is the numerator of the answer)

5/25 = 1/5

Eq 1: a = 2b

Eq 2: 3b = c

Eq 3: 2c = 3d

Rewrite Eq. 3 substituting using Eq. 2.

2(3b) = 3d (because c = 3b)

6b = 3d (simplify)

2b = d (divide by 3)

Since a and d both equal 2b, a = d. Therefore, d/a = 1.

If we were to multiply ab, bc, and ac together, we would get (ab)(bc)(ac) = a2b2c2 = (abc)2. If we were then to take the square-root of (abc)2, we would get abc, which is what the question asks us to find.

We know that ab = 4, bc = 5, and ac = 20. Thus (ab)(bc)(ac) = (4)(5)(20) = 400.

(ab)(bc)(ac) = a2b2c2 = 400.

(abc)2 = 400

abc = +√400 or –√400

abc = 20 or –20.

However, we are told that a, b, and c are all negative numbers, so the product of all three must be negative. Therefore, abc must be -20.

The answer is –20.

• 2√8 = 2√4 * √2 = 2 * 2√2 = 4√2
• 5√8 = 5√4 * √2 = 5 * 2√2 = 10√2
• 4√16 = 4 * 4 = 16

4√2 + 10√2 = 14√2

14√2 – 16

Begin by distributing each part:

2x(x2 + 4ax – 3a2) = 2x * x2 + 2x * 4ax – 2x * 3a2 = 2x3 + 8ax2 – 6a2x

The second:

–4a2(4x + 3a) = –16a2x – 12a3

Now, combine these:

2x3 + 8ax2 – 6a2x – 16a2x – 12a3

The only common terms are those with a2x; therefore, this reduces to

2x3 + 8ax2 – 22a2x – 12a3

This is the same as the given answer:

–12a3 – 22a2x + 8ax2 + 2x3

Substitute 4 for x + y in the expression given.

4 minus 6 equals –2.

Write an equation for the written expression: 9x – 6 = 48. When we solve for x we get x = 6.

If we substitute 6 in for x in the given equation and set our answer to 2, we can solve for y algebraically. 30 minus 4y divided by y equals 2 **-->**2y =30 -4y --> 6y =30 --> y=5. We could also work from the answers and substitute each answer in and solve.

To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x2 and -4 by 3.

x3 - 4x2 -12 + 15

You can then add -12 and 15, which equals 3.

You now have x3 - 4x2 + 3 and are finished. Just a reminder that x3 and 4x2 are not like terms as the x’s have different exponents.

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x2 + 2 = x2 + x – 2 – x2 + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8. Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

First, we can use the property of exponents that xy/xz = xy–z

Then we can use the property of exponents that states x–y = 1/xy

a–1b5c–1 = b5/ac

Cross multiply to get 14ay = 15bx, then divide by 15b to get x by itself.

Let f = Francine's age now.

e = f – 5

f = e + 5

In 2 years, Francine will be f + 2. Use our previous calculation to substitute.

f + 2 = (e + 5) + 2 = e + 7

Three consecutive positive integers are added together. If the largest of the three numbers is m, find the sum of the three numbers in terms of m.

If m is the largest of three consecutive positive integers, then the integers must be:

m – 2, m – 1, and m, where m > 2.

The sum of these three numbers is:

m - 2 + m – 1 + m = 3_m_ – 3

Using d = rt, we know that first part of the trip can be represented by f = rg. The second part of the trip can be represented by 30 = rx, where x is some unknown number of hours. Note that the rate r is in both equations because Sophie is traveling at the same rate as mentioned in the problem.

Solve each equation for the time (g in equation 1, x in equation 2).

g = f/r

x = 30/r

The total time is the sum of these two times

Note that, from equation 1, r = f/g, so

=

In order to translate words into symbols and algebraic expressions, we must take each piece of the statement one at a time. First, the problem mentions the square of the sum of p and q. This means we must first find the sum of p and q, and then find the square of this quantity. The sum of p and q can be represented by p + q. The square of this expression would be written as (p + q)2.

Where the sentence says "is equal to," we must put an equal sign. This means we can put an equal sign after (p + q)2. So our sentence is now as follows:

(p + q)2 =

Now, let's look at the last part of the sentence to see what should go on the right of the equal sign.

We must write an expression to represent the product of n and the positive difference of j and k. A product requires multiplication. So we are going to multiply n by the positive difference of j and k.

The difference of j and k could be represented as either j – k or as k – j. However, the statement calls for the positive difference. We don't know whether j or k is larger, so we don't know if j – k is positive, or if k – j is positive. The only way to ensure that we get a positive result is to take the absolute value of the difference. In other words, we can write the positive difference as either |j – k| or as |k – j|, because the absolute value will ensure that we yield a positive value. Also, because, in general, |a – b| = |b – a|, it doesn't matter whether or not we represent the positive difference of j and k as |j – k| or as |k – j|.

Now, let us finish writing the product of n and the positive difference of j and k. Because a product requires multiplication, we must multiply n by the positive difference of j and k, which we just determined can be either |j – k| or |k – j|. Thus, symbolically, we can write this as n • |j – k| or n • |k – j|.

Finally, when we put all of the pieces together, the equation becomes:

(p + q)2 = n • |k – j|

The answer is (p + q)2 = n • |k – j|.

Since y = –100t2 (in terms of t), we would like to replace the expression t with something in terms of x. Solving the first equation for t, we see:

x = 2_t_ + 3

x – 3 = 2_t_

(x – 3)/2 = t

Therefore, y = –100((_x – 3)/2)2 = –100(x – 3)2/4 = –25(x – 3)2