Algebra Formula

Algebra Formula with Example Algebraic Expressions in Maths

All Algebra formula

a2 – b2 = (a – b)(a + b)

(a + b)2 = a2 + 2ab + b2

a2 + b2 = (a + b)2 – 2ab

(a – b)2 = a2 – 2ab + b2

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca

(a + b)3 = a3 + 3a2b + 3ab2 + b3 

(a + b)3 = a3 + b3 + 3ab(a + b)

(a – b)3 = a3 – 3a2b + 3ab2 – b3 

(a – b)3 = a3 – b3 – 3ab(a – b)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4

a4 – b4 = (a – b)(a + b)(a2 + b2)

a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)

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Basic Algebra formulas

Linear Equation

Quadratic Formula

Distance Formula

Midpoint Formula

Slope Formula

Exponential Growth/Decay

Logarithmic Properties

Linear Equation

x=−bax = -\frac{b}{a}

Example 1.

Suppose you have the equation 2x + 3 = 7

To solve for x

2x+3=72x + 3 = 7

2x=7−32x = 7 – 3

2x=42x = 4

x=42=2x = \frac{4}{2} = 2

Example 2.

Solve for x in the equation 4x − 7 = 3x + 5

4x−7=3x+54x – 7 = 3x + 5

4x−3x=5+74x – 3x = 5 + 7

x=12x = 12

Quadratic Formula

ax2+bx+c=0ax^2 + bx + c = 0

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}

Example 1.

Consider the quadratic equation x2 – 4x + 4

Applying the Quadratic formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}

x=−(−4)±(−4)2−4(1)(4)2(1)x = \frac{-(-4) \pm \sqrt{(-4)^2 – 4(1)(4)}}{2(1)}

x=4±16−162x = \frac{4 \pm \sqrt{16 – 16}}{2}

x=4±02x = \frac{4 \pm \sqrt{0}}{2}

x=42=2x = \frac{4}{2} = 2

Distance Formula

The distance dd between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}

Example 1.

Given two points (1,3) and (−2,7), find the distance between them

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}

d=(−2−1)2+(7−3)2d = \sqrt{(-2 – 1)^2 + (7 – 3)^2}

d=9+16=25=5d = \sqrt{9 + 16} = \sqrt{25} = 5

Midpoint Formula

The midpoint MM between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Example 1.

For points (3,5) and (7,9), find the midpoint

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

M=(3+72,5+92)M = \left(\frac{3 + 7}{2}, \frac{5 + 9}{2}\right)

M=(5,7)M = (5, 7)

Slope Formula

The slope mm between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}

Example 1.

Find the slope between (−1,4) and (3,−2)

m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}

m=−2−43−(−1)m = \frac{-2 – 4}{3 – (-1)}

m=−64=−32m = \frac{-6}{4} = -\frac{3}{2}

Exponential Growth/Decay

The formula for exponential growth or decay is given by:

A=A0⋅(1±r)tA = A_0 \cdot (1 \pm r)^t

where AA is the final amount, A0A_0 is the initial amount, rr is the growth/decay rate, and tt is time.

Example 1.

If you have $1000 invested with an annual interest rate of 5%, find the amount after 3 years

A=A0⋅(1±r)tA = A_0 \cdot (1 \pm r)^t

A=1000⋅(1+0.05)3A = 1000 \cdot (1 + 0.05)^3

A=1000⋅(1.05)3≈1157.63A = 1000 \cdot (1.05)^3 \approx 1157.63

Logarithmic Properties

loga(xy)=loga(x)+loga(y)\log_a(xy) = \log_a(x) + \log_a(y)

loga(xy)=loga(x)−loga(y)\log_a\left(\frac{x}{y}\right) = \log_a(x) – \log_a(y)

loga(xn)=n⋅loga(x)\log_a(x^n) = n \cdot \log_a(x)

Example 1.

if ab=64a^b = 64, find loga64=b\log_a 64 = b:

loga64=b⟹ab=64\log_a 64 = b \implies a^b = 64

ab=a3⟹b=3a^b = a^3 \implies b = 3

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