Algebraic expressions

Correct answer: A. -1

Explanation: Substituting x=-3 gives 2x^2+5x-4=2(-3)^2+5(-3)-4=2(9)-15-4=-1. Therefore, the correct value is -1. Option 1 may result from a sign error; remember that (-3)^2=9. Exam tip: evaluate powers first, then perform multiplication and addition or subtraction.

Correct answer: B. (4a^2+3a+4)

Explanation: Direct answer: Option B, \\(4a^2+3a+4\\). To simplify a polynomial, combine only like terms: terms with the same variable and the same power. First collect the squared terms: \\(6a^2-2a^2=(6-2)a^2=4a^2\\). Next collect the terms containing a: \\(-4a+7a=(-4+7)a=3a\\). Finally combine constants: \\(9-5=4\\). Therefore the simplified expression is \\(4a^2+3a+4\\). Option A, \\(8a^2+3a+4\\), incorrectly adds the coefficients 6 and 2 instead of subtracting because the second squared term is negative. Option B is correct. Option C, \\(4a^2-11a+14\\), uses incorrect signs and combines unlike numerical parts. Option D, \\(8a^2-11a+4\\), has both the wrong squared-term coefficient and the wrong linear coefficient. The power of a term must not be changed while combining it. Memory cue: group by power: squared terms, first-power terms, then constants.

Correct answer: C. \(6x-10\)

Explanation: Subtracting \((2x+7)\) from \((8x-3)\) gives \((8x-3)-(2x+7)\). Since there is a minus sign before the second bracket, the signs of both its terms change: \(8x-3-2x-7=6x-10\). Hence, \(6x-10\) is correct. \(6x+4\) may result from incorrectly combining \(-3\) and \(+7\). Exam tip: When removing brackets after a minus sign, change the sign of every term inside the bracket.

Correct answer: C. 24

Explanation: Substituting the given values,
\(3x^2-2xy+y=3(2)^2-2(2)(-4)+(-4)=12+16-4=24\). Therefore, the correct answer is 24. Note that in
\(-2xy\), using
\(y=-4\) makes the product positive; treating it as
\(-16\) is a common error. In exams, substitute negative values in brackets to track signs correctly.

Correct answer: A. \(8m-17n\)

Explanation: Using the distributive property, \(5(2m-3n)=10m-15n\) and \(-2(m+n)=-2m-2n\). Therefore, \(10m-15n-2m-2n=(10m-2m)+(-15n-2n)=8m-17n\). In \(8m-13n\), the \(-2n\) term has not been combined correctly. Exam tip: When a negative coefficient is outside brackets, apply it to every term inside the brackets.

Correct answer: B. \(4(p+3)-p\)

Explanation: First, the sum of p and 3 is \(p+3\). Four times this entire sum is \(4(p+3)\). Subtracting p from it gives \(4(p+3)-p\). In option A, only p is multiplied by 4, not the whole sum. Exam tip: when a multiple of a sum is stated, enclose the sum in brackets.

Correct answer: C. -36

Explanation: Substitute r=-2: r^3=(-2)^3=-8 and r^2=(-2)^2=4. Therefore, r^3-4r^2+6r=-8-4(4)+6(-2)=-8-16-12=-36. Hence, option C is correct. A common error is taking r^2 as -4, but the square of a negative number is positive. Exam tip: use brackets while evaluating powers after substitution.

Correct answer: B. \(x-5\)

Explanation: First simplify the innermost bracket: \(4x-(2x-5)=4x-2x+5=2x+5\). Therefore, the complete expression becomes \(3x-(2x+5)=3x-2x-5=x-5\). Hence, \(x-5\) is correct. \(x+5\) results from incorrectly handling the minus sign before the outer bracket. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.

Correct answer: B. \(x-5\)

Explanation: First simplify the innermost grouped expression: \(4x-\{2x-5\}=4x-2x+5=2x+5\). Now, \(3x-[2x+5]=3x-2x-5=x-5\). Therefore, the correct answer is \(x-5\). The result \(x+5\) comes from not distributing the outer minus sign correctly to both terms inside the bracket. Exam tip: when a bracket is preceded by a minus sign, change the signs of all terms inside it while opening the bracket.

Correct answer: A. \(10ab+5a\)

Explanation: Combine like terms: \(7ab+3ab=10ab\) and \(-4a+9a=5a\). Hence, the simplified expression is \(10ab+5a\). The terms \(ab\) and \(a\) are unlike terms, so they cannot be combined. Exam tip: Add or subtract coefficients only when the variable parts of the terms are exactly the same.

Correct answer: A. -1

Explanation: Given (a+b)=9 and ab=14, substitute these directly into the expression: 3(a+b)-2ab=3(9)-2(14)=27-28=-1. The option 1 may result from an incorrect subtraction of 27-28. Exam tip: In such questions, there is no need to find a and b separately; directly use the given values of (a+b) and ab.

Correct answer: A. \(4x^2-9x\)

Explanation: Using the distributive property, \(2x(5x-3)=10x^2-6x\) and \(3x(2x+1)=6x^2+3x\). Hence, \((10x^2-6x)-(6x^2+3x)=4x^2-9x\). The option \(16x^2-9x\) results from incorrectly adding \(10x^2\) and \(6x^2\) instead of subtracting them. Exam tip: When a minus sign precedes a bracket, change the signs of every term inside that bracket.

Correct answer: A. \(-11x^2\)

Explanation: \((-3x^2)-(8x^2)=-3x^2-8x^2\). These are like terms because both have the variable part \(x^2\). Subtracting their coefficients gives \(-3-8=-11\), so the difference is \(-11x^2\). \(-11x^4\) is incorrect because subtraction of like terms does not change the exponent of \(x\). Exam tip: when adding or subtracting like terms, operate only on the coefficients.

Correct answer: A. 4

Explanation: On substituting \(x=4\), the numerator becomes \(4^2+2\times4-8=16+8-8=16\). Hence, \(\frac{16}{4}=4\), so option A is correct. Option C, \(6\), may result from an error while subtracting \(8\). Exam tip: For a fractional expression, evaluate the complete numerator before dividing by the denominator.

Correct answer: A. \(2x+8\)

Explanation: In the expression, \(-(2x-3)=-2x+3\) and \(4(x-1)=4x-4\). Therefore, \(9-2x+3+4x-4=2x+8\). Hence, the correct simplified form is \(2x+8\). The option \(6x+8\) can result from incorrectly taking \(-2x\) as \(+2x\). Exam tip: when a minus sign precedes brackets, change the sign of every term inside them.

Correct answer: B. \(6x+3\)

Explanation: The perimeter of a triangle is the sum of its three sides. Thus, \((2x+1)+(x+6)+(3x-4)=2x+x+3x+1+6-4=6x+3\). Hence, the correct expression is \(6x+3\). In \(6x+11\), the constant terms have been added incorrectly. In exams, combine like terms by adding the \(x\)-terms and constants separately.

Correct answer: A. (4p+2)

Explanation: Direct answer: Option A, \(4p+2\). Work from the innermost brackets outward. First, \(p-2\) is subtracted from \(3p\): \(3p-(p-2)=3p-p+2=2p+2\). Next calculate the square-bracket part: \(2p-(2p+2)=-2\). Finally, the original expression is \(4p-(-2)=4p+2\), because subtracting a negative number becomes addition. Option A is therefore correct. Option B, \(4p-2\), forgets that the bracketed value is negative and treats subtraction incorrectly. Option C, \(2p+2\), stops after an intermediate step and is not the complete answer. Option D, \(2p-2\), has both the wrong coefficient and wrong constant. The safest method is to simplify inside first and carefully distribute each minus sign.

Correct answer: B. 10

Explanation: Substituting the values, \(u^2v=(-1)^2\times3=3\), \(uv^2=(-1)\times3^2=-9\), and \(2u=2\times(-1)=-2\). Hence, \(u^2v-uv^2+2u=3-(-9)-2=3+9-2=10\). Therefore, 10 is correct. The value 12 would result from omitting the term \(2u=-2\). Exam tip: when subtracting a negative term, remember that \(-(-9)=+9\).

Correct answer: C. (x-4)^2

Explanation: The number 4 less than x is x-4. Since the question asks for the square of this entire number, the correct expression is (x-4)^2. The expression x^2-4 subtracts 4 only from the square of x; it is not the square of x-4. Exam tip: When a phrase asks for the square of a complete expression, place it in brackets before writing the exponent 2.

Correct answer: A. 4x^2+3xy

Explanation: Combine like terms: 7x^2 and -3x^2 add to 4x^2, while -2xy and 5xy add to 3xy. Therefore, the simplified form is 4x^2+3xy. The option 10x^2+3xy is incorrect because the coefficients of x^2 must be added as 7+(-3), not 7+3. Exam tip: Add or subtract only terms having the same variables with the same powers.

Correct answer: A. 34

Explanation: Given x-y=8. Taking 5 common from the first two terms of 5x-5y-6 gives 5(x-y)-6. Therefore, 5(8)-6=40-6=34. Hence, the correct answer is 34. The value 40 is only for 5(x-y); subtracting 6 is still necessary. Exam tip: Instead of finding x and y separately, directly substitute the given value of x-y.

Correct answer: A. \(4x+7y\)

Explanation: On expanding the brackets, \(3(2x-y)=6x-3y\) and \(-2(x-3y)=-2x+6y\). Therefore, the expression becomes \(6x-3y-2x+6y+4y=4x+7y\). The option \(4x+13y\) results from an incorrect combination of the \(y\)-terms. Exam tip: When multiplying a bracket by a negative number, apply the sign change to every term inside it.

Correct answer: A. 3x^2-2x-3

Explanation: Add terms with the same power of x: x^2+2x^2=3x^2, -3x+x=-2x, and 2-5=-3. Therefore, the sum is 3x^2-2x-3. In option C, the sign of the x-term is incorrect. Exam tip: While adding polynomials, group like terms first and then add their coefficients.

Correct answer: B. -9

Explanation: When x=0, the terms 7x^3, -5x^2, and 2x all become 0. Thus, the value of the expression is 0-0+0-9=-9. Zero is only the value of the variable terms; the constant term -9 must also be included. Exam tip: when x=0, only the constant term remains.

Correct answer: B. -4a-30

Explanation: First simplify the square bracket: 4a-2(a-5)=4a-2a+10=2a+10. Now, 2a-3(2a+10)=2a-6a-30=-4a-30. Therefore, -4a-30 is correct. In -4a+30, the sign of the constant term is incorrect because multiplying -3 by +10 gives -30. Exam tip: When a negative coefficient is outside brackets, multiply it by every term inside the bracket.