Equivalent Expressions I
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Advanced Math Domain Topics
Welcome to equivalent expressions in context part 1 by this point of the course, you've learned a lot of algebra tools Expanding factoring combining like terms exponent rules. This lesson is different. We're focusing on strategy Specifically, how do you recognize when a question can be solved much faster than the traditional algebraic approach now? Equivalent expression questions show up frequently on the SAT and they usually appear in one of two forms first which answer choice is equivalent Second find the missing constant inside a target form the good news Both question types often use the same exact shortcut before we get to that shortcut Let's talk about something important Different forms of the same expression reveal different information for example Factored form reveals zeros Vertex form reveals a maximum or minimum Expanded form reveals the y -intercept the SAT loves to hide the information you need in one form and present the expression in another Your job is to recognize when changing forms or using a shortcut Makes the answer easier to find Now for the shortcut, one of the most useful strategies for equivalent expression questions is plug in a value Choose a small value like two or three Try to avoid using zero and one because those values can sometimes make different expressions look equivalent Once you've chosen your numbers you evaluate the original expression and each answer choice If two expressions are truly equivalent they'll produce the same result and Remember your Desmos calculator is sitting right there Let it handle the arithmetic Let's try one Which expression is equivalent to the quantity x squared plus 10 squared?
plus the quantity x minus 4 times x plus 4 Before doing any algebra ask yourself could plugging in a value be faster Pause for a moment to think about it Okay, we're gonna see how fast this is with plugging in I'm gonna use x equals 2 and of course Desmos or some other calculator is doing the arithmetic for me plugging x equals 2 into the original expression yields 184 now, we just need to check the answer choices to find the match Only B also produces 184 so the answer is B.
That's why one well -chosen value can often eliminate three answer choices immediately one quick note Occasionally two answer choices can tie after you plug in your first value if that happens Don't panic. Just plug in a second value a value like three or four will usually break the tie now Let's use the exact same strategy in a different looking question this time. We're not looking for an equivalent expression We're looking for a missing constant But the underlying idea is exactly the same We're gonna plug a small value into both sides set them equal to each other and then solve Here's our student produced response example 2x squared plus 9x minus 18 can be written as The quantity 2x minus 3 times the quantity x plus K find K Before doing any expansion ask yourself could plugging in a value make this easier I hope that you're all nodding your heads. Yes Let's see plugging in here.
We'll use x equals 2 In fact, I will always use 2 or 3 as a test value unless a limitation in the question won't allow it Here 2 is good to go with 2 The first expression becomes 2 times 4 plus 9 times 2 minus 18, which equals 8 the second expression becomes The quantity 2 times 2 minus 3 times the quantity 2 plus K That's going to equal 1 times the quantity 2 plus K Which is going to just equal 2 plus K Now set the results of the two expressions equal to each other 2 plus K equals 8 subtract 2 from both sides to isolate the K and K equals 6 we got here by working with two lines of algebra Instead of expanding and matching coefficients and that's really the lesson whenever you see equivalent expressions matching forms or missing constants inside an expression ask yourself Can I plug in a value?
It won't solve every problem, but when it works, it's often the fastest path available Alright, we've already arrived at the summary different forms revealed different information the SAT often hides the information you need in one form and presents the expression in another Plug in a small value like 2 or 3 when equivalent expressions are involved Plugging in is often much faster than expanding or factoring The same shortcut works for many find the constant questions that's the big strategic takeaway from part 1 and Already that's gonna do it for equivalent expressions in context part 1.
It's not really about algebra It's about recognizing when a shortcut can save you time in part 2 We'll look at situations where plugging in doesn't fit and some of the most common traps students encounter when working with equivalent expressions
Read full transcriptplus the quantity x minus 4 times x plus 4 Before doing any algebra ask yourself could plugging in a value be faster Pause for a moment to think about it Okay, we're gonna see how fast this is with plugging in I'm gonna use x equals 2 and of course Desmos or some other calculator is doing the arithmetic for me plugging x equals 2 into the original expression yields 184 now, we just need to check the answer choices to find the match Only B also produces 184 so the answer is B.
That's why one well -chosen value can often eliminate three answer choices immediately one quick note Occasionally two answer choices can tie after you plug in your first value if that happens Don't panic. Just plug in a second value a value like three or four will usually break the tie now Let's use the exact same strategy in a different looking question this time. We're not looking for an equivalent expression We're looking for a missing constant But the underlying idea is exactly the same We're gonna plug a small value into both sides set them equal to each other and then solve Here's our student produced response example 2x squared plus 9x minus 18 can be written as The quantity 2x minus 3 times the quantity x plus K find K Before doing any expansion ask yourself could plugging in a value make this easier I hope that you're all nodding your heads. Yes Let's see plugging in here.
We'll use x equals 2 In fact, I will always use 2 or 3 as a test value unless a limitation in the question won't allow it Here 2 is good to go with 2 The first expression becomes 2 times 4 plus 9 times 2 minus 18, which equals 8 the second expression becomes The quantity 2 times 2 minus 3 times the quantity 2 plus K That's going to equal 1 times the quantity 2 plus K Which is going to just equal 2 plus K Now set the results of the two expressions equal to each other 2 plus K equals 8 subtract 2 from both sides to isolate the K and K equals 6 we got here by working with two lines of algebra Instead of expanding and matching coefficients and that's really the lesson whenever you see equivalent expressions matching forms or missing constants inside an expression ask yourself Can I plug in a value?
It won't solve every problem, but when it works, it's often the fastest path available Alright, we've already arrived at the summary different forms revealed different information the SAT often hides the information you need in one form and presents the expression in another Plug in a small value like 2 or 3 when equivalent expressions are involved Plugging in is often much faster than expanding or factoring The same shortcut works for many find the constant questions that's the big strategic takeaway from part 1 and Already that's gonna do it for equivalent expressions in context part 1.
It's not really about algebra It's about recognizing when a shortcut can save you time in part 2 We'll look at situations where plugging in doesn't fit and some of the most common traps students encounter when working with equivalent expressions