Equivalent Expressions - SAT Math

Which expression is equivalent to $(2x-3)(x+5)$ ?
Answer: $2x^2+7x-15$ To expand $(2x-3)(x+5)$ , we use FOIL or distribution method. First terms: $2x cdot x = 2x^2$ ; Outer terms: $2x cdot 5 = 10x$ ; Inner terms: $-3 cdot x = -3x$ ; Last terms: $-3 cdot 5 = -15$ . Combining these gives $2x^2+10x-3x-15 = 2x^2+7x-15$ . A common sign error occurs when multiplying the inner terms, forgetting that $-3 \times x = -3x$ , which would incorrectly lead to $2x^2+13x-15$ . When using FOIL, carefully track the sign of each term throughout the multiplication.
Which of the following expressions is equivalent to $(3x - 2)(x + 5)$ ?
Answer: $3x^2 + 13x - 10$ Expanding gives $3x^2 + 15x - 2x - 10 = 3x^2 + 13x - 10$ . The other choices come from sign or coefficient mistakes when distributing or combining like terms.
Which of the following expressions is equivalent to $\frac{2x^2-8}{2x}$ for $x\ne 0$ ?
Answer: $x-\frac{4}{x}$ Divide each term by $2x$ to get $x-\frac{4}{x}$ . The other choices reflect sign errors or incomplete simplification.
Expand and simplify the expression $(x-7)(x+3)$ . Which option matches the correct expansion after using FOIL and combining like terms?
Answer: $x^2-4x-21$ We need to expand $(x-7)(x+3)$ using the FOIL method. First terms: $x cdot x = x^2$ ; Outer terms: $x cdot 3 = 3x$ ; Inner terms: $-7 cdot x = -7x$ ; Last terms: $-7 cdot 3 = -21$ . Combining these: $x^2 + 3x - 7x - 21 = x^2 - 4x - 21$ . The key error to avoid is getting the sign wrong on the last term: $-7 cdot 3 = -21$ , not $+21$ . Always check your middle term by adding the outer and inner products carefully.
Which of the following expressions is equivalent to $\frac{x^2-9}{x-3}$ for $x \ne 3$ ?
Answer: $x+3$ Factor the numerator as $(x-3)(x+3)$ and cancel $x-3$ to get $x+3$ . The other choices come from subtracting incorrectly or not factoring the difference of squares.
Which of the following expressions is equivalent to $3(x-2)+2(x+5)$ ?
Answer: $5x+4$ Distribute and combine like terms: $3x-6+2x+10=5x+4$ . The other choices result from sign mistakes, failing to distribute, or incorrect constant addition.
Expand and simplify $(3t-2)(2t+5)$ . Which option is equivalent after using FOIL and combining like terms?
Answer: $6t^2+11t-10$ To expand $(3t-2)(2t+5)$ using FOIL: First: $3t cdot 2t = 6t^2$ ; Outer: $3t cdot 5 = 15t$ ; Inner: $-2 cdot 2t = -4t$ ; Last: $-2 cdot 5 = -10$ . Combining: $6t^2 + 15t - 4t - 10 = 6t^2 + 11t - 10$ . The key is correctly handling the middle terms: $15t - 4t = 11t$ , and getting the sign right on the constant: $-2 cdot 5 = -10$ . A common error is adding the middle terms incorrectly or writing the last term as $+10$ instead of $-10$ . Always double-check signs when multiplying negative numbers.
Simplify $\dfrac{6x^2y}{3xy^2}$ . Your answer should have no negative exponents and should be simplified by canceling common factors and reducing coefficients.
Answer: $\dfrac{2x}{y}$ To simplify $\frac{6x^2y}{3xy^2}$ , we divide both the coefficients and the variables. For coefficients: $\frac{6}{3} = 2$ . For variables: $\frac{x^2}{x} = x^{2-1} = x$ and $\frac{y}{y^2} = y^{1-2} = y^{-1} = \frac{1}{y}$ . Combining these: $\frac{6x^2y}{3xy^2} = 2 \cdot x \cdot \frac{1}{y} = \frac{2x}{y}$ . A common error is to cancel incorrectly, perhaps getting $\frac{2x^2}{y^2}$ by not properly subtracting exponents. Remember that when dividing powers with the same base, you subtract exponents: $\frac{x^m}{x^n} = x^{m-n}$ .
The expression $(3x-4)^2$ is used in a model and needs to be expanded. Which expression is equivalent to $(3x-4)^2$ after expanding completely?
Answer: $9x^2-24x+16$ The question requires expanding ( $(3x - 4)^2$ ) fully for a model. Use the formula ( $(a - b)^2$ = $a^2$ - 2ab + $b^2$ ), where a = 3x and b = 4. So, ( $(3x)^2$ = $9x^2$ ), (-2 cdot 3x cdot 4 = -24x), and ( $4^2$ = 16), resulting in ( $9x^2$ - 24x + 16). Verify by FOIL: 3x cdot 3x = $9x^2$ , 3x cdot (-4) = -12x, -4 cdot 3x = -12x, -4 cdot (-4) = 16, combining to $9x^2$ - 24x + 16. A key error is making the middle term positive, like +24x, by forgetting the negative in the expansion. Another mistake is treating it as a difference of squares without the middle term, like $9x^2$ - 16. For squaring binomials, always include the doubled cross term and check signs carefully.
Which of the following expressions is equivalent to $2x^2-5x-12$ ?
Answer: $(2x+3)(x-4)$ $(2x+3)(x-4)=2x^2-8x+3x-12=2x^2-5x-12$ . The other products yield incorrect middle terms or constants.
Which of the following expressions is equivalent to $2(3x - 5) - (x + 4) + 3$ ?
Answer: $5x - 11$ Distribute and combine like terms: $6x - 10 - x - 4 + 3 = 5x - 11$ . The distractors come from distributing errors or misadding constants.
Simplify $5(2y-3)-(y-4)+2y$ . Distribute carefully, note that subtracting $(y-4)$ changes both signs inside the parentheses, and then combine like terms to write the result as $ay+b$ .
Answer: $11y-11$ We need to simplify $5(2y-3)-(y-4)+2y$ by carefully distributing and combining like terms. First, distribute: $5(2y-3) = 10y-15$ . For $-(y-4)$ , the negative sign changes both terms inside: $-(y-4) = -y+4$ . Now we have $10y-15-y+4+2y$ . Combining like terms: $y$ terms give $10y-y+2y = 11y$ , and constants give $-15+4 = -11$ . Therefore, the expression equals $11y-11$ . The key error to avoid is writing $-(y-4)$ as $-y-4$ instead of $-y+4$ .
Simplify the expression $5y-2(3y-4)+7$ . Which expression is the simplified result?
Answer: $-y+15$ This question asks to simplify (5y - 2(3y - 4) + 7) by distributing and combining like terms. Begin by distributing the -2: (-2 cdot 3y = -6y) and (-2 cdot (-4) = 8), so the expression becomes (5y - 6y + 8 + 7). Now combine like terms: (5y - 6y = -y) for the y-terms, and (8 + 7 = 15) for the constants, resulting in (-y + 15). A common error is forgetting the positive 8 from distributing the negative to -4, leading to incorrect constants like +7 or -1. Another mistake involves not distributing the negative sign properly, such as treating it as subtraction without flipping the signs inside. When dealing with negative distributions, double-check the signs by expanding step-by-step to ensure accuracy.
Factor $x^2+10x+25$ completely.
Answer: $(x+5)(x+5)$ To factor $x^2+10x+25$ , we should check if it's a perfect square trinomial. A perfect square has the form $a^2+2ab+b^2=(a+b)^2$ . Here, $x^2=(x)^2$ , $25=5^2$ , and the middle term $10x=2(x)(5)$ . Since all three conditions are met, this is indeed $(x+5)^2$ , which can also be written as $(x+5)(x+5)$ . A common mistake is to factor this as $(x+5)(x-5)$ , confusing it with the difference of squares pattern. Perfect square trinomials always have the same sign in both factors, and the middle term helps determine whether it's $(a+b)^2$ or $(a-b)^2$ .
Which expression is equivalent to $9m^2-25$ ?
Answer: $(3m-5)(3m+5)$ To factor $9m^2-25$ , recognize this as a difference of squares: $a^2 - b^2 = (a-b)(a+b)$ . Here, $9m^2 = (3m)^2$ and $25 = 5^2$ , so $9m^2 - 25 = (3m)^2 - 5^2 = (3m-5)(3m+5)$ . A common mistake is trying to factor this as $(9m-5)(m+5)$ or not recognizing that 9 is a perfect square. Remember to check if both terms are perfect squares before attempting other factoring methods.
Which of the following expressions is equivalent to $\dfrac{x^2 - 9}{x^2 - 6x + 9}$ ?
Answer: $\dfrac{x + 3}{x - 3}$ Factor to get $\dfrac{(x-3)(x+3)}{(x-3)^2} = \dfrac{x+3}{x-3}$ (for $x \ne 3$ ). Other choices result from incorrect cancellation.
Which expression is equivalent to $6x^2 - 7x - 3$ , written as a product of two binomials?
Answer: $(2x - 3)(3x + 1)$ Factoring gives $(2x - 3)(3x + 1)$ , which expands to $6x^2 - 7x - 3$ . The other options produce incorrect middle coefficients or signs when expanded.
Which of the following is equivalent to $\frac{x^2 - 9}{x^2 - 6x + 9}$ ?
Answer: $\frac{x + 3}{x - 3}$ Factor: $x^2 - 9 = (x - 3)(x + 3)$ and $x^2 - 6x + 9 = (x - 3)^2$ . After canceling a factor of $(x - 3)$ , the result is $\frac{x + 3}{x - 3}$ (for $x \neq 3$ ).
Which of the following expressions is equivalent to $3(x+4) - 2x$ ?
Answer: $x+12$ Distribute to get $3x+12-2x=x+12$ . The other choices result from incorrect distribution or combining like terms.
Which expression is equivalent to $\dfrac{x^2-9}{x-3}$ for $x\ne 3$ ?
Answer: $x+3$ To simplify $\frac{x^2-9}{x-3}$ for $x≠3$ , we first factor the numerator. Recognizing that $x^2-9$ is a difference of squares, we can write it as $(x-3)(x+3)$ . Now we have $\frac{(x-3)(x+3)}{x-3}$ . Since $x≠3$ , we can cancel the common factor $(x-3)$ from both numerator and denominator, leaving us with $x+3$ . A common error is to try to divide term by term, getting something like $\frac{x^2}{x}-\frac{9}{-3}$ , which is algebraically incorrect. Always factor completely before canceling, and remember that you can only cancel factors, not terms.

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