√ 4 ⋅ √ 5 = √ 20 = √ 2*2*5 = 2 √ 5
Start with the numbers inside the parentheses. 4(| − 3 − 2|) − 5 4(| − 5|) − 5 Remember that the two vertical lines around “- 3 - 2” mean “absolute value.” The absolute value of -5 is 5, so: 4⋅5 − 5 = 15
Following the “Order of Operations”, first multiply 1⁄6 by −2 to get −2⁄6 so it becomes: −2⁄3 + (−2⁄6) Reduce 2⁄6 to 1⁄ and you have: −2⁄3 + (−1⁄3) = −1
There are several ways to approach a test item like this one. If the same numbers are given in each answer choice, simply convert them all to the same format. In this case, fractions and mixed numbers with the denominator of 6 would work well. Also note the simplification of the absolute value amount to −2⁄3, before changing to −4⁄6. (An absolute value is just the positive value of any number, so the negative of that would be negative.) 7⁄2 = 3 3⁄6 -5⁄6 = -5⁄6 −∣−2⁄3∣ = −4⁄6 1⁄2 = 4⁄8 = 3⁄6 Now, list them in the correct order and compare your listing with the answer choices to find the correct one. −5⁄6 < −4⁄6 < 3⁄6 < 3 3⁄6 For a test item in which the numbers in each of the answer choices are not the same, try to find errors that immediately “stick out” to you. For instance, in the choice −5⁄6 <1⁄2 < −∣−2⁄3∣ < 4⁄8 < 7⁄2, you’ll immediately notice that 1⁄2 is listed as less than 48 and you know those are equal, so that choice is wrong. There are pretty obvious inaccuracies in the other two incorrect choices for this problem, as well. So, you’d be able to narrow it down to only one possible choice, which you would need to check with the first process, above, to be sure.
4(−7+5) − (−2)(6−11) In a problem such as this one, it is necessary to follow the order of operations, PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Perform these steps, from left to right, in this order. So, performing the operations inside parentheses first: 4(−7+5) − (−2)(6−11) 4⋅(−2) − (−2)⋅(−5) Then, since there are no exponents, do multiplication next. −8 − 10 −18 PEMDAS can be remembered with the pneumonic, “Please Excuse My Dear Aunt Sally.”
Distribute x by multiplying with the two terms inside the parentheses: x(x2−3)=x3−3x. Then, combine with the other terms: x3−3x−2x2+5. Finally, arrange terms in such a way that the exponents of the variable x are in descending order with the constant as the last term. Thus: x3−2x2−3x+5.
Write the expression out as (5x+2y)(5x+2y). Then, multiply the expressions together by FOILing (multiple the First terms, the Outer terms, the Inner terms, and the Last terms). First terms: (5x)⋅(5x) = 25x2 Outer terms: (5x)⋅(2y)=10xy Inner terms: (2y)⋅(5x)=10xy Last terms: (2y)⋅(2y)=4y2 Combining all terms: 25x2+10xy+10xy+4y2 Add similar terms: 25x2+20xy+4y2
Factor the numerator using the reverse FOIL method: x2+8x+15=(x+5)(x+3). The (x+3) term on both the numerator and the denominator cancels out and we are left with (x+5), the answer.
Rearrange the equation so only the term with the x (in this equation, the ‘xy’ term) is on the left side. Thus: xy= 2y2−5y+4 Then, divide both sides by y to get an expression for x. The equation becomes: x=2y−5+4⁄y = 2y+4⁄y
Plug in the given values for x and y. Substituting: 3y2−4x+2xy 3(−1)2−4(6)+2(6)(−1) Perform the operations, from left to right: 3−24+(−12) −33
Plug in the given value y. Then rearrange terms so that only x is on the left side. 9x−12+y2=5 9x−12+32=5 9x−12+9=5 9x=12−9+5=8 x= 8⁄9
Find the roots of the polynomial by setting each expression equal to 0 and solving for x. For the expression (x+1) = 0: x+1 = 0 x=−1 Next, do the expression (x2−x−6)=0 x2−x−6 = 0 Factoring out: (x+2)(x−3) = 0 x = −2 and x = 3. Therefore, the roots are -1, -2 and 3
Start with the numbers inside the parentheses and then proceed with multiplication and division. (-3)(2+6)⁄4-2 (-3)(8)⁄4-2 = -24⁄4-2 = -6-2 = -8
To solve (x+3)(x2+3x−5), multiply x by all of the terms in the second expression, and then multiply 3 by all of the terms in the second expression. Then, add to combine like terms, arranging the exponents in descending order. (x+3)(x2+3x−5)=(x3+3x2−5x)+(3x2+9x−15)=x3+6x2+4x−15
Only the equation x2−9 = 0 holds true for the values of x=3 and x=-3. (3)2−9=9−9=0 and (−3)2−9=9−9=0
Add the different terms (which are all in fraction form) by making sure they have the same denominator. Multiply the first term by 4⁄4 and the second term by 2⁄2 so all 3 terms have 4x as denominator. 3Y⁄x - y⁄2X + 2y⁄4x 12y⁄4x - 2y⁄4x + 2y⁄4x (12y-2y+2y)⁄4x) = 12y⁄4x Simplifying, 12y⁄4x = 3y⁄x
5⁄2 must be the greatest because it is the only given fraction that is greater than one. Next must be 3⁄4 because it is the only remaining positive fraction. Finally, −1⁄4 is closer to 0 than −1⁄2 is, so −1⁄4 is greater than −1⁄2.
Plug in x = -1 into all terms. This gives us the numbers: -3⁄(2)(-1) = 3⁄2 1⁄(2)(-1) = -1⁄2 2(-1) = -2 From the greatest to least, the order would be: 3⁄2, 2⁄3, -1⁄2, -2 or -3⁄2x > 2⁄3 > 1⁄2x > 2x
Our goal is to isolate the x. 3 - 7x⁄2 < 10 3 - 3 - 7x⁄2 < 10 - 3 (-2⁄7 ⋅ − 7x⁄2 > 7 ⋅ (-2⁄7) x>−2. Flip the inequality sign because we divided by a negative number.
Start with the second equation and isolate x: x−2y=5 x=5+2y Substitute this result to x in the first equation: 3(5+2y)+4y = 25 15+6y+4y = 25 10y = 25−15 10y = 10 y = 10⁄10 = 1 Then, plug y = 1 into either equation (preferably the simpler one): x = 5+2(1) = 7
To factor the given polynomial: x2−x−12 Think of 2 numbers which will result to -12 when multiplied, and will result to -1 when added. (x+3)(x−4). Take note that: 3x-4 = -12, and 3+(-4) = -1. Either of (x+3) and (x−4) are factors, but only (x+3) is given in the choices; hence, it is the correct answer.
If pens cost $0.45 and pencils cost $0.15, the total cost of his purchase is represented by the equation: TotalPurchase=$0.45x+$0.15y This is not given as an option. However, it should be noticed that both 0.45 and 0.15 are divisible by 3, meaning a 3 can be factored out of both terms: 3($0.15x+$0.05y) which when expanded gives: $0.45x+$0.15y. Therefore, it is the correct answer.
The roots of a polynomial equation are the values that when substituted into the equation yield a true statement. ‘Finding roots’ is another way of describing, ‘solve for x.’ Note: The roots are not the same as the factors. The factors are the values that can be multiplied together to equal the original equation. You will find the factors first, then set each of them equal to 0 to find the roots. Finding the factors: x2+2x−15 = 0 Think of 2 numbers that result in -15 when multiplied, and result in 2 when added. That gives us the factors: (x−3)(x+5) = 0 Take note that: −3⋅5 = −15, and −3+5 = 2. Now, find the roots. Equate each factor to 0: x−3 = 0 x = 3 x+5 = 0 x = −5 The roots of the expression are 3 and -5.
To get the polynomial with 7 and -4 as roots, Use FOIL method and multiply the factors: (x−7)(x+4) = x2−3x−28 Hence, the polynomial x2−3x−28 has the roots 7 and -4.
This is a problem on factoring a difference of squares because there is an x2 term, a subtraction sign, no x term, then a constant. Start with: (x+_)(x−_) The two numbers will be the same but opposite in sign, and will be the square root of 16. Hence: (x+4)(x−4) These are the factors of the given expression.
To solve this word problem, set up a system of equations with the given information. Let q = the number of quarters and d = the number of dimes. She has 12 coins total, so we know that: q+d = 12 (equation 1) q = 12−d We also know the total value of the coins is $2.25, so we have: (equation 2) 0.25q+0.1d = 2.25 Plug in q from equation 1 into equation 2: 0.25(12−d)+0.1d = 2.25 3−0.25d+0.1d = 2.25 −0.15d = 2.25−3 d = −0.75−0.15 = 5 12−5 = 7 Therefore, there are 5 dimes and 7 quarters.
et up a system of equations with the given information. Let the 2 numbers be represented by x and y. Write “the sum of the two numbers is 27” as: x+y = 27 x = 27−y (equation 1) Write “3 times the first number is equal to 3 less than the second number” as: 3x = y−3 (equation 2) Solve equation 2 by using the value of x from equation 1: 3(27−y) = y−3 81−3y = y−3 −4y = −84 4y = 84 y = 21 x = 27−21 = 6
To factor the given polynomial x2+3x−40 Think of 2 numbers that result in -40 when multiplied, and result in 3 when added. (x−5)(x+8)
The total rental cost for the car is the rental for 5 days + the rental in excess of 500 miles Total rental cost = $78(5)+$0.20(563−500) = $390+$12.60 = $402.60
Try the given choices by plugging the values into the given equation. We can check each: For x = 0: 20 ≠ 02 For x = 1: 21 ≠ 12 For x = 2: 22 = 22 For x = 3: 23 ≠ 32 For x = 4: 24 = 42 For x = 5: 25 ≠ 52 So, “2 and 4” is the correct solution because all the other solutions contain at least one wrong answer.
√ 12 √ 3 can be simplified as √ 2*2*3 √ 3 = 2 √ 3 √ 3 = 2(3) = 6 Each of the following are equal to 6: √ 36 = 6 √ 9 √ 4 = 3*2 = 6 1⁄2 √ 144 = 1⁄2 * 12 = 6 Only 3 √ 9 = 3*3 = 9, hence, not equal to 6 or √ 12 √ 3
The problem says that y is the number of employees in the company. It starts with 25 people. At the start: y = 25 For every year (x), 10 employees are added: y = 25+10x Write the expression in the normal format, that is, terms with variable first, then the constant: y = 10x+25
The formula for volume is Volume = length⋅width⋅depth. W = 50feet L = 50+50%(50) =50+25 = 75feet D = 15(50) = 10feet Volume = LWD = (50)(75)(10) = 37,500feet3
Just like when multiplying two binomials, each term in the first set of parentheses must be multiplied by all of the terms in the second set. (x+2)(x2−3x+5)=x(x2−3x+5)+2(x2−3x+5)=(x3−3x2+5x)+(2x2−6x+10)=(x3−x2−x+10)
The formula for the circumference of a circle is: C=2πr. It is given in this question that: C=10π Equating C=2πr=10π r=10π2π=5 The formula for the area of a circle is A=πr2. Plug in r=5: A=π(52)=25π.
The formula for the area of a triangle is Area = 1⁄2 ⋅ Base ⋅ Height. Plug in the given values: 35in2 = 1⁄2 ⋅ Base ⋅ 7in Base = (35)(2)⁄7 = 10 The question asks for half of the length of the base, so the correct answer is 5inches.
Emily’s total revenue (R) equals total chocolate shakes sold (C) at $2.75 plus total vanilla shakes sold (V) at $2.50. Hence: R=$2.75C+$2.50V
