Algebraic expressions

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.

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.

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.

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.

\((-3x^2)-(8x^2)=-3x^2-8x^2\). 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.

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.

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.

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.

Inside (3p-(p-2)=2p+2), then (2p-(2p+2)=-2), so (4p-(-2)=4p+2). Simplify from inside to outside.

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 \(-(-9)=+9\).

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.

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.

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.

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.

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.

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.

The governing concept is the distributive law and careful handling of nested brackets. First simplify the innermost expression: 2(a - 5) = 2a - 10. Therefore, 4a - 2(a - 5) = 4a - (2a - 10) = 2a + 10. Now substitute this result into the full expression: 2a - 3(2a + 10). Distribute -3 across both terms: -3(2a + 10) = -6a - 30. Hence, 2a - 6a - 30 = -4a - 30. Therefore, option A is correct. Option B results from changing the sign of the constant incorrectly; options C and D fail to apply the negative multiplier to the variable term.

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.

Substituting m=3 and n=-2, m²n=3²×(-2)=9×(-2)=-18, while mn²=3×(-2)²=3×4=12. Therefore, m²n-mn²=-18-12=-30. Option -6 is not correct because it does not correctly subtract the second term, 12. Exam tip: the square of a negative number is positive, so (-2)²=4.

First simplify the innermost grouping: \(3x-\{x-4\}=3x-x+4=2x+4\). Then \(5(x+2)-[2x+4]=5x+10-2x-4=3x+6\). Therefore, the correct option is \(3x+6\). The result \(3x+14\) can occur if the minus sign before the bracket is not distributed correctly. Exam tip: when removing a bracket preceded by a minus sign, change the signs of all terms inside it.