Grammar
eśli dostanie tę pracę, przeprowadzi się do Krakowa i kupi tam mieszkanie.
[If she gets the job, she will move to Kraków and buy a flat there.]
Her solution was so ___ that what most of the class needed a full page to explain, she managed in just three lines.
concise/short
Please ___ ___ your homework to the teacher before you leave the classroom today.
hand in
Żeby wejść do muzeum, trzeba mieć bilet, ale dzieci poniżej siedmiu lat nie muszą za niego płacić.
[You must have a ticket to enter the museum, but children under seven don’t have to pay.]
3² + 4² = 5², and 5² − 3² = 4²
three squared plus four squared equals five squared, and five squared minus three squared equals four squared
Mężczyzna, którego telefon znalazłem w autobusie, następnego dnia zgłosił się na policję.
[The man whose phone I found on the bus went to the police the next day.]
A natural number greater than one that has no divisors other than one and itself is called a ___ number.
PRIME
It took me almost an hour to ___ ___ the value of x, because the equation had three unknowns.
solve for
work out
On na pewno jeszcze śpi: żaluzje są wciąż opuszczone, a w kuchni nie pali się światło.
[He must still be asleep; the blinds are still down, and the kitchen light isn’t on.]
(2/3)x³ − (5/7)x² + 1 = 0
two thirds x cubed minus five sevenths x squared plus one equals zero
Gdyby nie korki, dotarlibyśmy na lotnisko na czas.
[Had it not been for the traffic, we would have arrived at the airport on time.]
The line that touches a curve at exactly one point, without crossing it there, is called a ___. Its slope is equal to the derivative of the function at that point.
tangent
We checked our calculations several times, but they still don’t ___ ___: there is a gap of three units that we can’t explain.
add up
Powinieneś był mi wcześniej powiedzieć o terminie! Zacząłbym już w zeszłym tygodniu.
[You should have told me about the deadline earlier! I would have started last week.]
f′(x) = 2x, and f″(x) = 2 for all real x
f prime of x equals two x, and f double prime of x equals two for all real x
Nie tylko nie odpowiedział na mój e-mail, ale też nawet nie przeprosił za spóźnienie.
[Not only did he fail to reply to my email, but he also didn’t apologise for being late.]
As n grows larger, the terms of the sequence get closer and closer to zero, so we say that the sequence ___ to zero.
converges
She was brilliant at ___ ___ difficult ideas to beginners, usually by sketching a simple diagram on the board.
getting across
breaking down
Mogłeś mi powiedzieć, że nie przyjdziesz! Czekałem na ciebie przez godzinę.
[You could have told me you weren’t coming! I waited for you for an hour.]
x = (−b ± √(b² − 4ac)) / 2a
The solutions of a quadratic equation are given by minus b, plus or minus the square root of b squared minus four a c, all divided by two a. If the expression under the root is negative, there are no real solutions.
Zanim w piątek dotrzemy na miejsce, będziemy w drodze już od dziesięciu godzin.
[By the time we arrive on Friday, we will have been travelling for ten hours.]
The proof looked convincing at first, but it contained a subtle ___: in the third step it silently assumed the very statement it was trying to prove.
fallacy
When the professor _____ ___ my objection, I decided to ___ ___ my claim with data from three independent studies.
brushed aside
back up
Złodziej na pewno znał kod do alarmu, bo niemożliwe, żeby się włamał siłą: drzwi były nienaruszone.
[The thief must have known the alarm code, because he can’t have broken in by force; the door was undamaged.]
(sin x)′ = cos x, (ln x)′ = 1/x; ∫ 1/x dx = ln|x| + C
The derivative of the sine function is the cosine function, and the derivative of the natural logarithm of x is one over x. The integral of one over x equals the natural logarithm of the absolute value of x, plus a constant.