Algebra is a branch of mathematics that deals with symbols, variables, and the rules for manipulating them. Here are some important algebraic formulas, along with examples and their applications in daily life:

**Linear Equations:****Formula:**ax + b = c*a**x*+*b*=*c***Example:**2x + 3 = 72*x*+3=7**Application:**Linear equations are used in various real-life scenarios, such as calculating costs, determining distances, and solving problems involving rates of change.

**Quadratic Equations:****Formula:**ax^2 + bx + c = 0*a**x*2+*b**x*+*c*=0**Example:**2x^2 + 3x – 5 = 02*x*2+3*x*−5=0**Application:**Quadratic equations are used in physics to describe motion, in engineering for optimization problems, and in business for profit and loss analysis.

**Distance Formula:****Formula:**d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}*d*=(*x*2−*x*1)2+(*y*2−*y*1)2**Example:**Finding the distance between two points (x1, y1) and (x2, y2) on a coordinate plane.**Application:**It is used in navigation, surveying, and map-making to calculate distances between two locations.

**Pythagorean Theorem:****Formula:**a^2 + b^2 = c^2*a*2+*b*2=*c*2 (for a right triangle)**Example:**Finding the length of the hypotenuse of a right triangle when the lengths of the other two sides are known.**Application:**Used in construction, architecture, and engineering for designing structures and ensuring stability.

**Exponential Growth and Decay:****Formula:**A = P(1 + r)^t*A*=*P*(1+*r*)*t*(for exponential growth)**Example:**Calculating the future value of an investment with compound interest.**Application:**Used in finance to model investments, population growth, bacterial growth, and radioactive decay.

**Arithmetic and Geometric Sequences:****Formula:**- Arithmetic: a_n = a_1 + (n – 1)d
*a**n*=*a*1+(*n*−1)*d* - Geometric: a_n = a_1 \times r^{n-1}
*a**n*=*a*1×*r**n*−1

- Arithmetic: a_n = a_1 + (n – 1)d
**Example:**Finding the nth term of a sequence with a given first term (a1) and common difference (d) for arithmetic sequences, or common ratio (r) for geometric sequences.**Application:**Used in finance, computer algorithms, and natural phenomena modeling.

Understanding and applying these algebraic formulas can help solve a wide range of problems in daily life, from managing finances and planning routes to designing structures and analyzing data.

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