SAT Math: Equivalent expressions
Equivalent expressions: expanding, factoring and rational expressions.
- Advanced Math · Equivalent expressions
- Both modules
- 15 practice questions
- Calculator allowed (Desmos)
What the test covers
- Expanding, factoring, rational expressions, exponent rules
- Matching coefficients of identities
Key ideas
- \((a\pm b)^2=a^2\pm2ab+b^2\) and \(a^2-b^2=(a-b)(a+b)\).
- Two polynomials are equal for all \(x\) exactly when their coefficients match, term by term.
- Simplify a rational expression by factorising the top and bottom and cancelling common factors (for the allowed values of \(x\)).
- Exponent rules: \(x^a\cdot x^b=x^{a+b}\), \(\dfrac{x^a}{x^b}=x^{a-b}\), \(x^{1/n}=\sqrt[n]{x}\).
Common mistakes
- \((3x-2)^2\ne9x^2+4\): don't forget the middle term.
- Subtracting a product: bracket it first, then change every sign.
Do it in Desmos
Graph the original expression and each option; the equivalent one draws exactly the same curve.
Worked example
Worked example
Which expression is equivalent to \((3x-2)^2-(x+4)(x-4)\)?
- \(8x^2-12x-12\)
- \(10x^2-12x-12\)
- \(8x^2-12x+20\)
- \(8x^2+20\)
Answer: C: \(8x^2-12x+20\)
\((3x-2)^2=9x^2-12x+4\) and \((x+4)(x-4)=x^2-16\). The difference is \(8x^2-12x+20\).
Practice questions
Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.
Question 1
Which expression is equivalent to \(\dfrac{x^2-9}{x^2+x-6}\) for \(x\ne-3\) and \(x\ne2\)?
- \(\frac{x-3}{x+2}\)
- \(\frac{3}{2}\)
- \(\frac{x+3}{x+2}\)
- \(\frac{x-3}{x-2}\)
Show the answer and solution
Answer: D: \(\frac{x-3}{x-2}\)
\(\dfrac{(x-3)(x+3)}{(x+3)(x-2)}=\dfrac{x-3}{x-2}\).
Question 2
For all values of \(x\), \((2x+a)(x-3)=2x^2+bx-15\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -1
Expanding: \(2x^2+(a-6)x-3a\). Matching constants, \(-3a=-15\), so \(a=5\); then \(b=a-6=-1\).
Question 3
For \(x>0\), which expression is equivalent to \(\dfrac{x^{3/4}\cdot x^{1/2}}{x}\)?
- \(x^{\frac{5}{4}}\)
- \(x^{\frac{3}{8}}\)
- \(x^{-\frac{1}{4}}\)
- \(x^{\frac{1}{4}}\)
Show the answer and solution
Answer: D: \(x^{\frac{1}{4}}\)
\(x^{3/4+1/2-1}=x^{1/4}\).
Question 4
Which expression is equivalent to \(4x^2-25\)?
- \((4x-5)(x+5)\)
- \((4x+5)(x-5)\)
- \((2x-5)^2\)
- \((2x-5)(2x+5)\)
Show the answer and solution
Answer: D: \((2x-5)(2x+5)\)
Difference of two squares: \((2x)^2-5^2=(2x-5)(2x+5)\).
Question 5
Which expression is equivalent to \((2x-5)^2-(x+4)(x-4)\)?
- \(3 x^{2} - 20 x + 9\)
- \(3 x^{2} - 28 x + 9\)
- \(3 x^{2} + 41\)
- \(3 x^{2} - 20 x + 41\)
Show the answer and solution
Answer: D: \(3 x^{2} - 20 x + 41\)
\((2x-5)^2=4 x^{2} - 20 x + 25\) and \((x+4)(x-4)=x^2-16\). The difference is \(3 x^{2} - 20 x + 41\).
Question 6
Which expression is equivalent to \((3x-2)^2-(x+2)(x-2)\)?
- \(8 x^{2} - 16 x\)
- \(8 x^{2} - 12 x\)
- \(8 x^{2} - 12 x + 8\)
- \(8 x^{2} + 8\)
Show the answer and solution
Answer: C: \(8 x^{2} - 12 x + 8\)
\((3x-2)^2=9 x^{2} - 12 x + 4\) and \((x+2)(x-2)=x^2-4\). The difference is \(8 x^{2} - 12 x + 8\).
Question 7
Which expression is equivalent to \((4x-2)^2-(x+3)(x-3)\)?
- \(15 x^{2} - 16 x - 5\)
- \(15 x^{2} - 16 x + 13\)
- \(15 x^{2} + 13\)
- \(15 x^{2} - 22 x - 5\)
Show the answer and solution
Answer: B: \(15 x^{2} - 16 x + 13\)
\((4x-2)^2=16 x^{2} - 16 x + 4\) and \((x+3)(x-3)=x^2-9\). The difference is \(15 x^{2} - 16 x + 13\).
Question 8
Which expression is equivalent to \((4x-5)^2-(x+4)(x-4)\)?
- \(15 x^{2} - 48 x + 9\)
- \(15 x^{2} + 41\)
- \(15 x^{2} - 40 x + 41\)
- \(15 x^{2} - 40 x + 9\)
Show the answer and solution
Answer: C: \(15 x^{2} - 40 x + 41\)
\((4x-5)^2=16 x^{2} - 40 x + 25\) and \((x+4)(x-4)=x^2-16\). The difference is \(15 x^{2} - 40 x + 41\).
Question 9
Which expression is equivalent to \((2x-5)^2-(x+3)(x-3)\)?
- \(3 x^{2} - 26 x + 16\)
- \(3 x^{2} - 20 x + 34\)
- \(3 x^{2} + 34\)
- \(3 x^{2} - 20 x + 16\)
Show the answer and solution
Answer: B: \(3 x^{2} - 20 x + 34\)
\((2x-5)^2=4 x^{2} - 20 x + 25\) and \((x+3)(x-3)=x^2-9\). The difference is \(3 x^{2} - 20 x + 34\).
Question 10
For all values of \(x\), \((4x+a)(x-3)=4x^2+bx-9\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -9
Expanding: \(4x^2+(a-12)x-3a\). Matching constants, \(-3a=-9\), so \(a=3\); then \(b=a-12=-9\).
Question 11
For all values of \(x\), \((2x+a)(x-4)=2x^2+bx-4\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -7
Expanding: \(2x^2+(a-8)x-4a\). Matching constants, \(-4a=-4\), so \(a=1\); then \(b=a-8=-7\).
Question 12
For all values of \(x\), \((4x+a)(x-4)=4x^2+bx-4\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -15
Expanding: \(4x^2+(a-16)x-4a\). Matching constants, \(-4a=-4\), so \(a=1\); then \(b=a-16=-15\).
Question 13
For all values of \(x\), \((3x+a)(x-3)=3x^2+bx-6\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -7
Expanding: \(3x^2+(a-9)x-3a\). Matching constants, \(-3a=-6\), so \(a=2\); then \(b=a-9=-7\).
Question 14
For all values of \(x\), \((4x+a)(x-2)=4x^2+bx-2\), where \(a\) and \(b\) are constants. What is the value of \(b\)?
Show the answer and solution
Answer: -7
Expanding: \(4x^2+(a-8)x-2a\). Matching constants, \(-2a=-2\), so \(a=1\); then \(b=a-8=-7\).
Keep going
- Previous topic: Linear inequalities
- Next topic: Nonlinear equations and systems
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