TACHS Quiz: Simplifying And Evaluating Expressions
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Simplifying And Evaluating ExpressionsQuestion 1 of 13

Simplify 5x−105+3x5\dfrac{5x - 10}{5} + \dfrac{3x}{5}.

2x5−2\dfrac{2x}{5} - 2
8x5+2\dfrac{8x}{5} + 2
8x5−2\dfrac{8x}{5} - 2
2x5+2\dfrac{2x}{5} + 2
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TACHS Quiz

TACHS Quiz: Simplifying And Evaluating Expressions

Practice Simplifying And Evaluating Expressions in TACHS with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Simplifying And Evaluating Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for TACHS.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Simplify 5x−105+3x5\dfrac{5x - 10}{5} + \dfrac{3x}{5}.

  1. 2x5−2\dfrac{2x}{5} - 2
  2. 8x5+2\dfrac{8x}{5} + 2
  3. 8x5−2\dfrac{8x}{5} - 2 (correct answer)
  4. 2x5+2\dfrac{2x}{5} + 2
Explanation: When you see fractions with the same denominator, you can combine them by adding or subtracting the numerators while keeping the denominator the same. This problem tests your ability to distribute and combine like terms within fraction addition. Let's work through this step by step. First, notice that both fractions have the same denominator of 5, so we can combine them: 5x−105+3x5=(5x−10)+3x5\dfrac{5x - 10}{5} + \dfrac{3x}{5} = \dfrac{(5x - 10) + 3x}{5} Now simplify the numerator by combining like terms: (5x−10)+3x=5x+3x−10=8x−10(5x - 10) + 3x = 5x + 3x - 10 = 8x - 10 So our expression becomes: 8x−105\dfrac{8x - 10}{5} We can split this into separate terms: 8x5−105=8x5−2\dfrac{8x}{5} - \dfrac{10}{5} = \dfrac{8x}{5} - 2 This matches answer choice C. Let's examine why the other choices are wrong. Choice A gives 2x5−2\dfrac{2x}{5} - 2, which incorrectly combines the x-terms as 5x+3x=2x5x + 3x = 2x instead of 8x8x. Choice B shows 8x5+2\dfrac{8x}{5} + 2, which gets the x-term right but makes the constant positive instead of negative—forgetting that −10÷5=−2-10 ÷ 5 = -2. Choice D combines both errors: wrong x-coefficient and wrong sign on the constant. Remember: when adding fractions with the same denominator, combine the numerators carefully and watch your signs. Always double-check your arithmetic, especially when distributing negative signs and combining like terms.

Question 2

Simplify 5 (2y−3)−2 (y−1)5\,(2y - 3) - 2\,(y - 1).

  1. 10y−510y - 5
  2. 8y−138y - 13 (correct answer)
  3. 3y−13y - 1
  4. 8y+138y + 13
Explanation: This problem tests your ability to distribute multiplication over parentheses and combine like terms—two fundamental algebraic skills that often appear together. Start by distributing each coefficient to the terms inside the parentheses. For the first part, 5(2y−3)5(2y - 3), multiply 5 by each term: 5×2y=10y5 \times 2y = 10y and 5×(−3)=−155 \times (-3) = -15. For the second part, −2(y−1)-2(y - 1), multiply -2 by each term: −2×y=−2y-2 \times y = -2y and −2×(−1)=+2-2 \times (-1) = +2. This gives you 10y−15−2y+210y - 15 - 2y + 2. Now combine like terms by grouping the yy terms and the constant terms separately: (10y−2y)+(−15+2)=8y−13(10y - 2y) + (-15 + 2) = 8y - 13. This matches choice B. Choice A (10y−510y - 5) represents the error of only distributing the first term and incorrectly combining constants. Choice C (3y−13y - 1) occurs when you subtract the coefficients incorrectly (10−2=310 - 2 = 3 instead of 10−2=810 - 2 = 8) and make sign errors with the constants. Choice D (8y+138y + 13) gets the coefficient right but flips the sign on the constant term, likely from mishandling the negative distribution. When distributing negative signs, be extra careful with your arithmetic. Write out each step clearly: distribute first, then combine like terms. Double-check your signs—negative distributions are where most students make errors on problems like this.

Question 3

Simplify the expression 3x+4x−2+53x + 4x - 2 + 5.

  1. −7x+3-7x + 3
  2. 7x−37x - 3
  3. 7x+37x + 3 (correct answer)
  4. −7x−3-7x - 3
Explanation: When you see an algebraic expression with multiple terms, your goal is to combine like terms—terms that have the same variable part or are both constants. Let's work through 3x+4x−2+53x + 4x - 2 + 5 step by step. First, identify the like terms: 3x3x and 4x4x are like terms (both have the variable xx), while −2-2 and +5+5 are like terms (both are constants with no variables). Combine the xx terms: 3x+4x=7x3x + 4x = 7x. Think of this as having 3 of something plus 4 more of the same thing, giving you 7 total. Combine the constant terms: −2+5=3-2 + 5 = 3. You're starting at -2 on a number line and moving 5 positions to the right, landing at 3. Your final simplified expression is 7x+37x + 3, which is answer choice C. Now let's see where the wrong answers come from. Choice A (−7x+3-7x + 3) incorrectly makes the coefficient of xx negative—this happens if you mistakenly subtract instead of adding the xx terms. Choice B (7x−37x - 3) gets the xx term right but makes the constant negative, likely from calculating −2+5-2 + 5 as −3-3 instead of +3+3. Choice D (−7x−3-7x - 3) combines both errors, getting both the coefficient and constant wrong. Remember: when combining like terms, pay careful attention to the signs in front of each term. It helps to rewrite subtraction as adding a negative: 3x+4x+(−2)+53x + 4x + (-2) + 5.

Question 4

Let m=12m = \dfrac{1}{2}. What is 8m2−3m+28m^{2} - 3m + 2?

  1. 12\dfrac{1}{2}
  2. 72\dfrac{7}{2}
  3. 52\dfrac{5}{2} (correct answer)
  4. 114\dfrac{11}{4}
Explanation: When you encounter an expression with a given variable value, you need to substitute carefully and follow the order of operations. This tests your ability to work with fractions and evaluate algebraic expressions systematically. Let's substitute m=12m = \frac{1}{2} into 8m2−3m+28m^2 - 3m + 2. First, calculate m2=(12)2=14m^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}. Now substitute both values: 8m2−3m+2=8⋅14−3⋅12+28m^2 - 3m + 2 = 8 \cdot \frac{1}{4} - 3 \cdot \frac{1}{2} + 2 Simplify each term: 8⋅14=28 \cdot \frac{1}{4} = 2, 3⋅12=323 \cdot \frac{1}{2} = \frac{3}{2}, and the constant remains 22. So we have: 2−32+2=4−322 - \frac{3}{2} + 2 = 4 - \frac{3}{2} Convert to a common denominator: 4=824 = \frac{8}{2}, so 82−32=52\frac{8}{2} - \frac{3}{2} = \frac{5}{2}. The answer is C. Let's examine the wrong choices: Choice A (12\frac{1}{2}) likely results from only calculating one term or making multiple arithmetic errors. Choice B (72\frac{7}{2}) could come from incorrectly adding instead of subtracting the middle term, giving 4+32=1124 + \frac{3}{2} = \frac{11}{2}, then making another error. Choice D (114\frac{11}{4}) might result from calculation errors with the fractions or incorrect order of operations. Always substitute step-by-step, square fractions carefully (remember that (ab)2=a2b2\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}), and convert to common denominators before adding or subtracting fractions. Double-check your arithmetic with fraction operations.

Question 5

Evaluate (p−13)2\left(\dfrac{p-1}{3}\right)^{2} for p=7p = 7.

  1. 649\dfrac{64}{9}
  2. 1616
  3. 83\dfrac{8}{3}
  4. 44 (correct answer)
Explanation: This question tests your ability to substitute values into algebraic expressions and follow the order of operations carefully. To evaluate (p−13)2\left(\dfrac{p-1}{3}\right)^{2} when p=7p = 7, start by substituting the given value: (7−13)2\left(\dfrac{7-1}{3}\right)^{2}. Next, simplify inside the parentheses first. Calculate 7−1=67-1 = 6, giving you (63)2\left(\dfrac{6}{3}\right)^{2}. Then simplify the fraction: 63=2\dfrac{6}{3} = 2, so you have (2)2(2)^{2}. Finally, square the result: 22=42^{2} = 4. The answer is D. Let's examine why the other choices are incorrect. Choice A (649\dfrac{64}{9}) results from a common error where students square both the numerator and denominator separately without simplifying first, getting (63)2=6232=369=4\left(\dfrac{6}{3}\right)^{2} = \dfrac{6^{2}}{3^{2}} = \dfrac{36}{9} = 4. Wait—that's still 4. Actually, choice A comes from squaring 6 and 3 separately to get 369\dfrac{36}{9}, then making an arithmetic error. Choice B (16) happens when students correctly find that p−1=6p-1 = 6, but then square 6 before dividing by 3, getting 363=12\dfrac{36}{3} = 12—though that's not 16 either. Choice B likely comes from squaring 4 instead of 2. Choice C (83\dfrac{8}{3}) results from forgetting to square the entire fraction and instead just doubling it. Remember: when evaluating expressions with substitution, always work step-by-step through the order of operations. Simplify completely before applying exponents to avoid fraction arithmetic errors.

Question 6

Which expression is equivalent to 2(x+4)−(3x−5)2(x + 4) - (3x - 5)?

  1. 5x−135x - 13
  2. −x+13-x + 13 (correct answer)
  3. x−13x - 13
  4. −x−13-x - 13
Explanation: When you see an expression with parentheses and subtraction like this, you need to carefully distribute and combine like terms. The key is handling the negative sign in front of the second set of parentheses correctly. Let's work through 2(x+4)−(3x−5)2(x + 4) - (3x - 5) step by step. First, distribute the 2 into the first parentheses: 2(x+4)=2x+82(x + 4) = 2x + 8. Next, handle the subtraction of the second expression. When you subtract (3x−5)(3x - 5), you're distributing a negative sign to each term inside: −(3x−5)=−3x+5-(3x - 5) = -3x + 5. Notice that subtracting a negative 5 becomes positive 5. Now combine: 2x+8−3x+52x + 8 - 3x + 5. Group like terms: (2x−3x)+(8+5)=−x+13(2x - 3x) + (8 + 5) = -x + 13. Choice A (5x−135x - 13) likely comes from incorrectly adding the x-coefficients (2 + 3 = 5) and subtracting the constants incorrectly. Choice C (x−13x - 13) suggests adding the x-coefficients as 2 + 1 = 3, then somehow getting just x, while also getting the wrong constant. Choice D (−x−13-x - 13) correctly finds the x-term but makes the same constant error as choices A and C, probably from calculating 8−5=38 - 5 = 3 instead of 8+5=138 + 5 = 13. The correct answer is B: −x+13-x + 13. Study tip: When subtracting expressions in parentheses, rewrite the problem by distributing the negative sign first. This prevents sign errors that are extremely common on standardized tests.

Question 7

For t=−2t=-2, find t2−4tt\dfrac{t^{2}-4t}{t}.

  1. 1212
  2. 66
  3. −12-12
  4. −6-6 (correct answer)
Explanation: This problem tests algebraic substitution and simplification with negative numbers. When you see an algebraic expression to evaluate at a specific value, substitute carefully and follow the order of operations. Let's substitute t=−2t = -2 into t2−4tt\frac{t^2 - 4t}{t}: (−2)2−4(−2)−2\frac{(-2)^2 - 4(-2)}{-2} First, evaluate the numerator: (−2)2=4(-2)^2 = 4 and 4(−2)=−84(-2) = -8, so the numerator becomes 4−(−8)=4+8=124 - (-8) = 4 + 8 = 12. Now we have 12−2=−6\frac{12}{-2} = -6. Answer choice A (1212) represents the value of just the numerator, forgetting to divide by t=−2t = -2. This is a common error when students calculate the pieces correctly but don't complete the final step. Answer choice B (66) comes from correctly finding the numerator as 1212 but then dividing by 22 instead of −2-2, losing track of the negative sign. Answer choice C (−12-12) results from getting the wrong sign in the numerator calculation, likely computing 4−4(−2)=4−8=−44 - 4(-2) = 4 - 8 = -4 incorrectly, then getting −4−2=2\frac{-4}{-2} = 2 and making additional errors, or from −12−2\frac{-12}{-2} if the numerator was miscalculated as −12-12. Answer choice D (−6-6) is correct. When substituting negative values, write them in parentheses to avoid sign errors. Also, consider simplifying algebraically first: t2−4tt=t(t−4)t=t−4\frac{t^2 - 4t}{t} = \frac{t(t-4)}{t} = t-4, then substitute to get −2−4=−6-2-4 = -6.

Question 8

Evaluate 2a2−3a+42a^{2} - 3a + 4 when a=−2a = -2.

  1. −2-2
  2. −18-18
  3. 1818 (correct answer)
  4. 00
Explanation: When you see an algebraic expression that needs to be evaluated at a specific value, you're being tested on substitution and order of operations. The key is to carefully replace the variable with the given value and follow PEMDAS. To evaluate 2a2−3a+42a^{2} - 3a + 4 when a=−2a = -2, substitute −2-2 for every aa: 2(−2)2−3(−2)+42(-2)^{2} - 3(-2) + 4 Now follow the order of operations. First, handle the exponent: (−2)2=4(-2)^{2} = 4 (remember that a negative number squared becomes positive). This gives you: 2(4)−3(−2)+42(4) - 3(-2) + 4 Next, perform the multiplications: 2(4)=82(4) = 8 and −3(−2)=6-3(-2) = 6 (negative times negative equals positive). You now have: 8+6+48 + 6 + 4 Finally, add from left to right: 8+6+4=188 + 6 + 4 = 18 Looking at the wrong answers: Choice A (−2-2) likely comes from incorrectly calculating (−2)2=−4(-2)^{2} = -4 instead of 44. Choice B (−18-18) probably results from sign errors, such as getting −3(−2)=−6-3(-2) = -6 instead of 66. Choice D (00) might occur if you make multiple computational mistakes that coincidentally cancel out. The most common trap here is mishandling negative numbers, especially with exponents and multiplication. Remember: (−2)2=4(-2)^{2} = 4, but −22=−4-2^{2} = -4. Always use parentheses when substituting negative values, and double-check your signs at each step.

Question 9

Simplify 13(9x−6)+4\dfrac{1}{3}(9x - 6) + 4.

  1. 3x2+4\dfrac{3x}{2} + 4
  2. 3x−23x - 2
  3. 3x+23x + 2 (correct answer)
  4. 6x+46x + 4
Explanation: This problem tests your ability to use the distributive property and combine like terms—two fundamental algebraic skills that appear frequently on standardized tests. To simplify 13(9x−6)+4\frac{1}{3}(9x - 6) + 4, start by distributing 13\frac{1}{3} to both terms inside the parentheses. Multiply 13×9x=3x\frac{1}{3} \times 9x = 3x and 13×(−6)=−2\frac{1}{3} \times (-6) = -2. This gives you 3x−2+43x - 2 + 4. Finally, combine the constant terms: −2+4=2-2 + 4 = 2. The simplified expression is 3x+23x + 2, which is answer choice C. Let's examine why the other options are incorrect. Choice A, 3x2+4\frac{3x}{2} + 4, results from incorrectly distributing 13\frac{1}{3} to only the 9x9x term, giving 9x3=3x\frac{9x}{3} = 3x but then mistakenly writing it as 3x2\frac{3x}{2}, and failing to distribute to the −6-6 at all. Choice B, 3x−23x - 2, correctly distributes 13\frac{1}{3} to get 3x−23x - 2, but then forgets to add the +4+4 term at the end. Choice D, 6x+46x + 4, appears to result from multiplying 9x9x by 23\frac{2}{3} instead of 13\frac{1}{3}, or from some other fundamental error in distribution. When simplifying expressions with distribution, work systematically: distribute first, then combine like terms. Double-check that you've applied the distributive property to every term inside the parentheses, and don't forget about terms outside the parentheses that need to be combined at the end.

Question 10

Simplify −3(2k−5)+4(k+3)-3(2k - 5) + 4(k + 3).

  1. −2k−27-2k - 27
  2. 2k+272k + 27
  3. −2k+27-2k + 27 (correct answer)
  4. 2k−272k - 27
Explanation: When you encounter an expression with multiple terms in parentheses, you need to apply the distributive property carefully to each part, then combine like terms. Start by distributing each coefficient to the terms inside its parentheses. For −3(2k−5)-3(2k - 5), multiply −3-3 by both 2k2k and −5-5: −3⋅2k=−6k-3 \cdot 2k = -6k and −3⋅(−5)=+15-3 \cdot (-5) = +15. This gives you −6k+15-6k + 15. For 4(k+3)4(k + 3), multiply 44 by both kk and 33: 4⋅k=4k4 \cdot k = 4k and 4⋅3=124 \cdot 3 = 12. This gives you 4k+124k + 12. Now your expression becomes: −6k+15+4k+12-6k + 15 + 4k + 12. Combine like terms by grouping the kk terms and the constants: (−6k+4k)+(15+12)=−2k+27(-6k + 4k) + (15 + 12) = -2k + 27. Looking at the wrong answers: Choice A (−2k−27-2k - 27) correctly finds the kk coefficient but makes a sign error with the constants—likely from miscalculating 15+1215 + 12 or the distribution. Choice B (2k+272k + 27) gets the constant term right but makes a sign error with the kk coefficient, probably from −6+4=2-6 + 4 = 2 instead of −2-2. Choice D (2k−272k - 27) combines both sign errors from choices A and B. The key strategy here is to work methodically: distribute first, then combine like terms. Double-check your signs at each step, especially when distributing negative coefficients—this is where most errors occur on these problems.

Question 11

If x=6x = 6, what is the value of 3x2−4\dfrac{3x}{2} - 4?

  1. −5-5
  2. 1313
  3. 72\dfrac{7}{2}
  4. 55 (correct answer)
Explanation: This question tests your ability to substitute values into algebraic expressions and follow the order of operations correctly. When you're given that x=6x = 6, you need to substitute this value into the expression 3x2−4\frac{3x}{2} - 4 and calculate step by step. First, substitute: 3(6)2−4\frac{3(6)}{2} - 4. Next, multiply in the numerator: 182−4\frac{18}{2} - 4. Then divide: 9−49 - 4. Finally, subtract: 55. Let's examine why each wrong answer appears. Choice A (−5-5) likely comes from making a sign error somewhere in the calculation, perhaps treating the subtraction incorrectly or making an arithmetic mistake that flips the sign. Choice B (1313) suggests you might have added 4 instead of subtracting it, getting 9+4=139 + 4 = 13. Choice C (72\frac{7}{2}) indicates you probably subtracted 4 from the numerator before dividing: 18−42=142=7\frac{18-4}{2} = \frac{14}{2} = 7, then mistakenly left it as 72\frac{7}{2} instead of converting to the whole number 7. This violates the order of operations since you must complete the division 182\frac{18}{2} before subtracting 4. The key strategy here is to always follow the order of operations (PEMDAS) carefully. When substituting values into expressions, work methodically through each step and double-check your arithmetic. Watch especially for the common trap of performing operations in the wrong sequence when fractions are involved.

Question 12

Simplify 4b2+3b−2b2−5b+14b^{2} + 3b - 2b^{2} - 5b + 1.

  1. 6b2−8b+16b^{2} - 8b + 1
  2. 2b2−2b+12b^{2} - 2b + 1 (correct answer)
  3. 2b2+8b−12b^{2} + 8b - 1
  4. −2b2−2b+1-2b^{2} - 2b + 1
Explanation: When you see an expression with multiple terms containing the same variable, your goal is to combine like terms by grouping terms with identical variable parts and adding their coefficients. To simplify 4b2+3b−2b2−5b+14b^{2} + 3b - 2b^{2} - 5b + 1, first identify and group the like terms. You have three types: b2b^{2} terms, bb terms, and constant terms. Rearranging: (4b2−2b2)+(3b−5b)+1(4b^{2} - 2b^{2}) + (3b - 5b) + 1. Now combine each group by adding the coefficients. For the b2b^{2} terms: 4−2=24 - 2 = 2, so you get 2b22b^{2}. For the bb terms: 3−5=−23 - 5 = -2, giving you −2b-2b. The constant remains 11. Your final answer is 2b2−2b+12b^{2} - 2b + 1. Choice A (6b2−8b+16b^{2} - 8b + 1) incorrectly adds the b2b^{2} coefficients as 4+2=64 + 2 = 6 instead of 4−2=24 - 2 = 2, and adds the bb coefficients as 3+5=83 + 5 = 8 instead of 3−5=−23 - 5 = -2. Choice C (2b2+8b−12b^{2} + 8b - 1) correctly finds the b2b^{2} term but makes the same sign errors as choice A for the bb term and constant. Choice D (−2b2−2b+1-2b^{2} - 2b + 1) gets the bb term and constant correct but incorrectly calculates the b2b^{2} coefficient as −2-2 instead of +2+2. Always pay careful attention to signs when combining like terms. Write out the operation explicitly (like 4−24 - 2) rather than doing it mentally to avoid sign errors.

Question 13

Simplify: 7−[2(3n−4)−(n−6)]7 - \bigl[2(3n - 4) - (n - 6)\bigr].

  1. 5n−95n - 9
  2. 5n+95n + 9
  3. −5n−9-5n - 9
  4. −5n+9-5n + 9 (correct answer)
Explanation: When you encounter expressions with nested brackets and parentheses, work systematically from the inside out, carefully tracking positive and negative signs throughout each step. Start by simplifying inside the brackets: 2(3n−4)−(n−6)2(3n - 4) - (n - 6). Distribute the 2: 2(3n−4)=6n−82(3n - 4) = 6n - 8. For the second term, distribute the negative sign: −(n−6)=−n+6-(n - 6) = -n + 6. Combining these gives: 6n−8−n+6=5n−26n - 8 - n + 6 = 5n - 2. Now substitute back into the original expression: 7−[5n−2]7 - [5n - 2]. Distribute the negative sign in front of the brackets: 7−5n+2=−5n+97 - 5n + 2 = -5n + 9. Choice A (5n−95n - 9) results from forgetting to distribute the negative sign in front of the brackets, keeping 5n5n positive instead of making it −5n-5n. Choice B (5n+95n + 9) makes the same sign error with the nn term but also incorrectly handles the constant terms. Choice C (−5n−9-5n - 9) correctly makes the nn term negative but fails to properly distribute the negative sign to the constant term −2-2, which should become +2+2. The correct answer is D: −5n+9-5n + 9. Remember that distributing a negative sign changes the sign of every term inside the brackets or parentheses. When simplifying complex expressions, work methodically through one set of grouping symbols at a time, and double-check your sign changes at each step.