Mathematics

Equations and sequences

Q: A quadratic equation step by step

Solve the equation x² − 5x + 6 = 0. Write down the coefficients: a = 1, b = −5, c = 6. Work out the discriminant: Δ = (−5)² − 4 · 1 · 6 = 25 − 24 = 1. The discriminant is positive, so there are two roots: √Δ = 1. Substitute into the formula: x₁ = (5 − 1) / 2 = 2, x₂ = (5 + 1) / 2 = 3. Check with Vieta's formulas: x₁ + x₂ = 5 = −b/a, x₁ · x₂ = 6 = c/a. Answer: x₁ = 2, x₂ = 3.

Q: A system of equations by determinants

Solve the system: 2x + 3y = 12 and 4x − y = 10. Main determinant: W = 2 · (−1) − 4 · 3 = −14. Determinant for x: Wx = 12 · (−1) − 10 · 3 = −42. Determinant for y: Wy = 2 · 10 − 4 · 12 = −28. Divide: x = −42 / −14 = 3, y = −28 / −14 = 2. Check it in the first equation: 2 · 3 + 3 · 2 = 12. Answer: x = 3, y = 2.

Linear and quadratic equations, systems of two equations, arithmetic and geometric sequences.

A quadratic equation is solved through the discriminant, whose sign tells you in advance how many roots there will be. A system of two equations is solved by determinants, which shows immediately whether the system has a single solution. The sequence tools give any term you name and the sum.

Calculators

Linear equation

ax + b = 0 ⇒ x = −b / a

Coefficient a

Constant term b

Root3
Value at x = 1-5
Slope2.5

Quadratic equation

ax² + bx + c = 0, Δ = b² − 4ac

Coefficient a

Coefficient b

Constant term c

Roots2 ; 3

Discriminant1
Sum of the roots5
Product of the roots6
Vertex(2.5 ; -0.25)

System of two equations

a₁x + b₁y = c₁ ; a₂x + b₂y = c₂

a₁

b₁

c₁

a₂

b₂

c₂

Solutionx = 3 ; y = 2

Determinant W-14
Determinant Wx-42
Determinant Wy-28

Arithmetic sequence

aₙ = a₁ + (n − 1) · r

First term a₁

Common difference r

Term number n

Term number 2098

Sum of n terms1,010
Middle term50.5
Hundredth term498

Geometric sequence

aₙ = a₁ · qⁿ⁻¹

First term a₁

Common ratio q

Term number n

Term number 1076.8867

Sum of n terms226.6602
Infinite sumdivergent
Growth per term50.00%

Tables and cheat sheets

Algebraic identities

Formula	Expansion	Example
(a + b)²	a² + 2ab + b²	(x + 3)² = x² + 6x + 9
(a − b)²	a² − 2ab + b²	(x − 5)² = x² − 10x + 25
a² − b²	(a − b)(a + b)	x² − 49 = (x − 7)(x + 7)
(a + b)³	a³ + 3a²b + 3ab² + b³	(x + 1)³ = x³ + 3x² + 3x + 1
a³ − b³	(a − b)(a² + ab + b²)	x³ − 8 = (x − 2)(x² + 2x + 4)
a³ + b³	(a + b)(a² − ab + b²)	x³ + 27 = (x + 3)(x² − 3x + 9)

The discriminant and Vieta's formulas

for the equation ax² + bx + c = 0

Condition	Roots	Note
Δ > 0	x = (−b ± √Δ) / 2a	two distinct roots
Δ = 0	x₀ = −b / 2a	one double root
Δ < 0	no real roots	the parabola does not cross the x-axis
sum of the roots	x₁ + x₂ = −b / a	Vieta's formula
product of the roots	x₁ · x₂ = c / a	Vieta's formula

Worked examples with full solutions

A quadratic equation step by step

Solve the equation x² − 5x + 6 = 0.

Write down the coefficientsa = 1, b = −5, c = 6
Work out the discriminantΔ = (−5)² − 4 · 1 · 6 = 25 − 24 = 1
The discriminant is positive, so there are two roots√Δ = 1
Substitute into the formulax₁ = (5 − 1) / 2 = 2, x₂ = (5 + 1) / 2 = 3
Check with Vieta's formulasx₁ + x₂ = 5 = −b/a, x₁ · x₂ = 6 = c/a

Answer: x₁ = 2, x₂ = 3.

A system of equations by determinants

Solve the system: 2x + 3y = 12 and 4x − y = 10.

Main determinantW = 2 · (−1) − 4 · 3 = −14
Determinant for xWx = 12 · (−1) − 10 · 3 = −42
Determinant for yWy = 2 · 10 − 4 · 12 = −28
Dividex = −42 / −14 = 3, y = −28 / −14 = 2
Check it in the first equation2 · 3 + 3 · 2 = 12

Answer: x = 3, y = 2.