Equations, Fractions and Parabola Formulas

A broad collection of algebra essentials: motion, work and cost problems, fractions and proportions, GCD and LCM, linear and quadratic equations, the discriminant, Vieta's formulas and the vertex of a parabola. Use the calculator under each formula to check your own solutions.

37 formulas · Grades 5–9

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All 37 formulas on this page
  1. Distance formula
  2. Speed formula
  3. Time of motion formula
  4. Speed downstream
  5. Speed upstream
  6. Cost of a purchase
  7. Work (productivity)
  8. Division with remainder
  9. Arithmetic mean
  10. Fundamental property of a fraction
  11. Multiplying common fractions
  12. Dividing common fractions
  13. Fundamental property of a proportion
  14. Absolute value of a number
  15. Adding numbers with different signs
  16. Multiplying negative numbers
  17. Multiplying numbers with different signs
  18. GCD (greatest common divisor)
  19. LCM (least common multiple)
  20. Adding fractions with the same denominator
  21. Subtracting fractions with the same denominator
  22. Finding an unknown term of a proportion
  23. Map scale
  24. Linear function
  25. Linear equation (root)
  26. System of two linear equations
  27. Discriminant
  28. Roots of a quadratic equation
  29. Vieta's formulas (reduced form)
  30. Factoring a quadratic trinomial
  31. Vieta's formulas (general case)
  32. Inverse proportionality function
  33. Vertex of a parabola (x)
  34. Vertex of a parabola (y)
  35. Median of a sample (5 numbers)
  36. Mode of a sample (5 numbers)
  37. Range of a sample (5 numbers)

Distance formula

Grade 5
S=v⋅tS = v \cdot t

Distance travelled during uniform motion.

Legend:
  • v — Speed (v)
  • t — Time (t)

Speed formula

Grade 5
v=Stv = \frac{S}{t}

Speed as the ratio of distance to time.

Legend:
  • S — Distance (S)
  • t — Time (t)

Time of motion formula

Grade 5
t=Svt = \frac{S}{v}

Time of motion as the ratio of distance to speed.

Legend:
  • S — Distance (S)
  • v — Speed (v)

Speed downstream

Grade 5
v=v0+v1v = v_0 + v_1

Sum of the object's own speed and the current's speed.

Legend:
  • v_self — Own speed
  • v_flow — Current speed

Speed upstream

Grade 5
v=v0−v1v = v_0 - v_1

Difference between the object's own speed and the current's speed.

Legend:
  • v_self — Own speed
  • v_flow — Current speed

Cost of a purchase

Grade 5
C=a⋅nC = a \cdot n

Total cost as the price per unit multiplied by the quantity.

Legend:
  • a — Unit price (a)
  • n — Count (n)

Work (productivity)

Grade 5
A=v⋅tA = v \cdot t

Amount of work done, expressed through rate and time.

Legend:
  • v — Rate of work (v)
  • t — Time (t)

Division with remainder

Grade 5
a=b⋅q+ra = b \cdot q + r

Dividend expressed through the divisor, quotient and remainder.

Legend:
  • b — Divisor (b)
  • q — Quotient (q)
  • r — Remainder (r)

Arithmetic mean

Grade 5
M=a1+a2+⋯+annM = \frac{a_1 + a_2 + \dots + a_n}{n}

Calculates the arithmetic mean of two or three given numbers.

Legend:
  • a1 — Number 1
  • a2 — Number 2
  • a3 — Number 3 (optional: 0)

Fundamental property of a fraction

Grade 6
ab=a⋅cb⋅c\frac{a}{b} = \frac{a \cdot c}{b \cdot c}

Multiplying the numerator and denominator by c keeps the fraction's value unchanged.

Legend:
  • a — Numerator a
  • b — Denominator b
  • c — Factor c (c≠0)

Multiplying common fractions

Grade 6
ab⋅cd=a⋅cb⋅d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

The numerators and denominators are multiplied separately.

Legend:
  • a — Numerator of the first fraction
  • b — Denominator of the first fraction
  • c — Numerator of the second fraction
  • d — Denominator of the second fraction

Dividing common fractions

Grade 6
ab:cd=a⋅db⋅c\frac{a}{b} : \frac{c}{d} = \frac{a \cdot d}{b \cdot c}

Multiplying the first fraction by the reciprocal of the second.

Legend:
  • a — Numerator of the first fraction
  • b — Denominator of the first fraction
  • c — Numerator of the second fraction
  • d — Denominator of the second fraction

Fundamental property of a proportion

Grade 6
ab=cd⇒a⋅d=b⋅c\frac{a}{b} = \frac{c}{d} \Rightarrow a \cdot d = b \cdot c

The product of the extreme terms of a proportion equals the product of the middle terms.

Legend:
  • a — Extreme term (1st)
  • b — Middle term (1st)
  • c — Middle term (2nd)
  • d — Extreme term (2nd)

Absolute value of a number

Grade 6
∣a∣={a,a≥0−a,a<0|a| = \begin{cases} a, & a \ge 0 \\ -a, & a < 0 \end{cases}

The absolute value of a number (its distance from zero on the number line).

Legend:
  • a — Number a

Adding numbers with different signs

Grade 6
a+(−b)=a−ba + (-b) = a - b

Adding a negative number is the same as subtracting its absolute value.

Legend:
  • a — Number a
  • b — Absolute value of negative b

Multiplying negative numbers

Grade 6
(−a)⋅(−b)=a⋅b(-a) \cdot (-b) = a \cdot b

The product of two negative numbers is always positive.

Legend:
  • a — Absolute value of the first number (a)
  • b — Absolute value of the second number (b)

Multiplying numbers with different signs

Grade 6
(−a)⋅b=−(a⋅b)(-a) \cdot b = -(a \cdot b)

The product of numbers with different signs is always negative.

Legend:
  • a — Absolute value of the first (a)
  • b — Absolute value of the second (b)

GCD (greatest common divisor)

Grade 6
gcd⁡(a,b)\gcd(a, b)

The largest number that divides both numbers with no remainder.

Legend:
  • a — Number a
  • b — Number b

LCM (least common multiple)

Grade 6
lcm⁡(a,b)=a⋅bgcd⁡(a,b)\operatorname{lcm}(a,b) = \frac{a \cdot b}{\gcd(a,b)}

The smallest number that is a multiple of both numbers.

Legend:
  • a — Number a
  • b — Number b

Adding fractions with the same denominator

Grade 6
ac+bc=a+bc\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}

The numerators are added, the denominator stays the same.

Legend:
  • a — Numerator a
  • b — Numerator b
  • c — Denominator c

Subtracting fractions with the same denominator

Grade 6
ac−bc=a−bc\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}

The numerators are subtracted, the denominator stays the same.

Legend:
  • a — Numerator a
  • b — Numerator b
  • c — Denominator c

Finding an unknown term of a proportion

Grade 6
ab=cx⇒x=b⋅ca\frac{a}{b}=\frac{c}{x} \Rightarrow x=\frac{b \cdot c}{a}

Finds the fourth (unknown) term of a proportion.

Legend:
  • a — First term of the proportion (a)
  • b — Second term of the proportion (b)
  • c — Third term of the proportion (c)

Map scale

Grade 6
M=DdM = \frac{D}{d}

Shows how many times the real-world distance is greater than the distance on the map.

Legend:
  • d — Distance on the map (d)
  • D — Real-world distance (D)

Linear function

Grade 7
y=kx+by = kx + b

Calculates the value of y from the argument x, slope k and offset b.

Legend:
  • k — Coefficient k
  • x — Argument x
  • b — Constant term b

Linear equation (root)

Grade 7
ax+b=0⇒x=−baax+b=0 \Rightarrow x=-\frac{b}{a}

Root of a linear equation in one variable.

Legend:
  • a — Coefficient a
  • b — Coefficient b

System of two linear equations

Grade 7
x=c1b2−c2b1a1b2−a2b1, y=a1c2−a2c1a1b2−a2b1x=\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1},\ y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}

Solving a system of two linear equations in two variables using determinants.

Legend:
  • a1 — Coefficient a1
  • b1 — Coefficient b1
  • c1 — Coefficient c1
  • a2 — Coefficient a2
  • b2 — Coefficient b2
  • c2 — Coefficient c2

Discriminant

Grade 8
D=b2−4acD = b^2 - 4ac

Determines the number of real roots of a quadratic equation ax² + bx + c = 0.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Roots of a quadratic equation

Grade 8
x1,2=−b±D2ax_{1,2} = \frac{-b \pm \sqrt{D}}{2a}

Finding the roots of an equation using the discriminant.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Vieta's formulas (reduced form)

Grade 8
x1+x2=−p,x1⋅x2=qx_1 + x_2 = -p, \quad x_1 \cdot x_2 = q

Relates the roots of x² + px + q = 0 to its coefficients.

Legend:
  • p — Coefficient p
  • q — Coefficient q

Factoring a quadratic trinomial

Grade 8
ax2+bx+c=a(x−x1)(x−x2)ax^2 + bx + c = a(x - x_1)(x - x_2)

Expressing a trinomial as a product using its roots.

Legend:
  • a — Coefficient a
  • x1 — Root x1
  • x2 — Root x2

Vieta's formulas (general case)

Grade 8
x1+x2=−ba,x1⋅x2=cax_1+x_2=-\frac{b}{a},\quad x_1 \cdot x_2=\frac{c}{a}

Relates the roots of the quadratic ax² + bx + c = 0 to its coefficients.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Inverse proportionality function

Grade 8
y=kxy = \frac{k}{x}

Value of the inverse-proportionality function at a point x.

Legend:
  • k — Coefficient k
  • x — Argument x

Vertex of a parabola (x)

Grade 9
x0=−b2ax_0 = -\frac{b}{2a}

Calculates the x-coordinate of the vertex of the parabola ax² + bx + c.

Legend:
  • a — Coefficient a
  • b — Coefficient b

Vertex of a parabola (y)

Grade 9
y0=ax02+bx0+cy_0 = a x_0^2 + b x_0 + c

Calculates the y-coordinate of the vertex from x₀ and the coefficients.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c
  • x0 — Coordinate x0

Median of a sample (5 numbers)

Grade 9
Me=x(3)\mathrm{Me} = x_{(3)}

The number in the middle of an ordered data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

Mode of a sample (5 numbers)

Grade 9
Mo\mathrm{Mo}

The value that occurs most often in a data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

Range of a sample (5 numbers)

Grade 9
R=xmax−xminR = x_{max} - x_{min}

Difference between the largest and smallest value in a data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

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