Algebra Formulas for SAT and ACT Math flashcards

Question 1

Expand (a + b)² for numbers a and b.

Accepted Answer

a² + 2ab + b²

Question 2

Expand (a − b)² for numbers a and b.

Accepted Answer

a² − 2ab + b²

Question 3

Factor the perfect square trinomial a² + 2ab + b².

Accepted Answer

(a + b)²

Question 4

Factor the perfect square trinomial a² − 2ab + b².

Accepted Answer

(a − b)²

Question 5

Factor the difference of squares a² − b².

Accepted Answer

(a − b)(a + b)

Question 6

Product of powers: simplify a^m · a^n (same base a, exponents m and n).

Accepted Answer

a^(m + n)

Question 7

Quotient of powers: simplify a^m / a^n (same nonzero base a, exponents m and n).

Accepted Answer

a^(m − n)

Question 8

Power of a power: simplify (a^m)^n (base a, exponents m and n).

Accepted Answer

a^(mn)

Question 9

Negative exponent: what is a^(−n) for a nonzero base a?

Accepted Answer

1 / a^n

Question 10

Write the slope-intercept form of a line's equation, with slope m and constant b.

Accepted Answer

y = mx + b

Question 11

Write the point-slope form of a line's equation through the point (x1, y1) with slope m.

Accepted Answer

y − y1 = m(x − x1)

Question 12

What is the slope m of the line through the points (x1, y1) and (x2, y2)?

Accepted Answer

m = (y2 − y1) / (x2 − x1)

Question 13

Write the vertex form of a quadratic function f(x) with leading coefficient a and constants h and k.

Accepted Answer

f(x) = a(x − h)² + k

Question 14

State the quadratic formula for the solutions x of ax² + bx + c = 0.

Accepted Answer

x = (−b ± √(b² − 4ac)) / (2a)

Question 15

What is the equation of the axis of symmetry of the parabola y = f(x) = ax² + bx + c?

Accepted Answer

x = −b / (2a)

Question 16

Factor the sum of cubes a³ + b³.

Accepted Answer

(a + b)(a² − ab + b²)

Question 17

Factor the difference of cubes a³ − b³.

Accepted Answer

(a − b)(a² + ab + b²)

Question 18

Arithmetic sequence with first term a1 and common difference d: what is the nth term a_n?

Accepted Answer

a_n = a1 + d(n − 1)

Question 19

Geometric sequence with first term a1 and common ratio r: what is the nth term a_n?

Accepted Answer

a_n = a1 · r^(n − 1)

Question 20

Arithmetic series: what is the sum S_n of the first n terms, using the first term a1 and nth term a_n?

Accepted Answer

S_n = n(a1 + a_n) / 2

Question 21

Geometric series with first term a1 and common ratio r ≠ 1: what is the sum S_n of the first n terms?

Accepted Answer

S_n = a1(1 − r^n) / (1 − r)

Question 22

In ax² + bx + c = 0, what is the discriminant D?

Accepted Answer

D = b² − 4ac

Question 23

Using the discriminant D = b² − 4ac, what are the roots of ax² + bx + c = 0?

Accepted Answer

(−b ± √D) / (2a)

Question 24

What is the sum of the two roots of ax² + bx + c = 0 (a ≠ 0)?

Accepted Answer

−b / a

Question 25

What is the product of the two roots of ax² + bx + c = 0 (a ≠ 0)?

Accepted Answer

c / a

Question 26

What is the distance d between the points (x1, y1) and (x2, y2) in the coordinate plane?

Accepted Answer

d = √((x2 − x1)² + (y2 − y1)²)

Question 27

What are the coordinates of the midpoint of the segment joining (x1, y1) and (x2, y2)?

Accepted Answer

((x1 + x2) / 2, (y1 + y2) / 2)

Question 28

Sum of an arithmetic progression: for the n terms a, a + d, a + 2d, …, a + (n − 1)d, what is the sum written as n·a plus a multiple of d?

Accepted Answer

n·a + n(n − 1)d / 2

Question 29

Sum of an arithmetic progression: for n terms with first term a and last term ℓ, what is the sum?

Accepted Answer

n(a + ℓ) / 2

Question 30

Geometric progression: what is a + ax + ax² + … + ax^(n − 1) for x ≠ 1?

Accepted Answer

a(1 − x^n) / (1 − x)

Question 31

What is b^0 for any nonzero base b?

Accepted Answer

1

Question 32

What is the natural logarithm of 1, ln(1)?

Accepted Answer

0

Question 33

What is ln(e), the natural logarithm of the base e?

Accepted Answer

1

Question 34

Change of base: write log_a(z) using natural logarithms (a and z positive, a ≠ 1).

Accepted Answer

log_a(z) = ln(z) / ln(a)

Question 35

Change of base: write log_a(z) using logarithms to a new base b (a, b, z positive; a, b ≠ 1).

Accepted Answer

log_a(z) = log_b(z) / log_b(a)

Question 36

What is log_a(b) in terms of log_b(a) (a, b positive and not 1)?

Accepted Answer

log_a(b) = 1 / log_b(a)

Question 37

Write e^(−x) in terms of e^x.

Accepted Answer

e^(−x) = 1 / e^x

Question 38

Product law: write ln(xy) for positive numbers x and y.

Accepted Answer

ln(xy) = ln(x) + ln(y)

Question 39

Quotient law: write ln(x/y) for positive numbers x and y.

Accepted Answer

ln(x/y) = ln(x) − ln(y)

Question 40

Power law: write ln(a^x) for a positive base a and real exponent x.

Accepted Answer

ln(a^x) = x · ln(a)

Question 41

Write ln(1/x) for a positive number x.

Accepted Answer

−ln(x)

Question 42

Inverse relation: what is e^(ln x) for positive x?

Accepted Answer

x

Question 43

Inverse relation: what is ln(e^x) for real x?

Accepted Answer

x

Question 44

Same base: simplify a^p · a^q for a nonzero base a.

Accepted Answer

a^(p + q)

Question 45

Same exponent: simplify a^p · b^p for positive bases a and b.

Accepted Answer

(ab)^p

Question 46

Simplify e^p · e^q.

Accepted Answer

e^(p + q)

Question 47

Simplify (e^p)^q for real p and q.

Accepted Answer

e^(pq)

Algebra Formulas for SAT and ACT Math flashcards

The cards

How this deck was checked

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Algebra Formulas for SAT and ACT Math flashcards

The cards

How this deck was checked

Sources

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