There are many formulas for algebra, but here are some of the most common and useful ones:
– **Basic operations**: To add, subtract, multiply or divide two algebraic expressions, we can use the distributive property and combine like terms. For example, (x + 2) + (3x – 4) = 4x – 2 and (x + 2) – (3x – 4) = -2x + 6. To multiply two expressions, we can use the FOIL method (First, Outer, Inner, Last) or the area model. For example, (x + 2)(3x – 4) = 3x^2 – 4x + 6x – 8 = 3x^2 + 2x – 8. To divide two expressions, we can use long division or synthetic division. For example, (3x^2 + 2x – 8) / (x + 2) = 3x – 4.
– **Exponents and radicals**: To simplify expressions with exponents and radicals, we can use the following rules:
* Product rule: a^m * a^n = a^(m+n)
* Quotient rule: a^m / a^n = a^(m-n)
* Power rule: (a^m)^n = a^(mn)
* Negative exponent rule: a^-n = 1 / a^n
* Zero exponent rule: a^0 = 1
* Fractional exponent rule: a^(m/n) = n√(a^m)
* Product to power rule: (ab)^n = a^n * b^n
* Quotient to power rule: (a/b)^n = a^n / b^n
* Power of power rule: (a^m)^n = a^(mn)
– **Factoring**: To factor an expression means to write it as a product of simpler expressions. There are different methods of factoring depending on the type of expression. Some common methods are:
* Common factor: If an expression has a common factor in all its terms, we can factor it out. For example, 6x^2 + 9x = 3x(2x + 3).
* Difference of squares: If an expression is of the form a^2 – b^2, we can factor it as (a + b)(a – b). For example, x^2 – 25 = (x + 5)(x – 5).
* Sum or difference of cubes: If an expression is of the form a^3 + b^3 or a^3 – b^3, we can factor it as (a + b)(a^2 – ab + b^2) or (a – b)(a^2 + ab + b^2), respectively. For example, x^3 + 8 = (x + 2)(x^2 – 2x + 4).
* Trinomial: If an expression is of the form ax^2 + bx + c, we can factor it by finding two numbers p and q such that pq = ac and p + q = b. Then we can write the expression as a(x + p/a)(x + q/a). For example, x^2 + 5x + 6 = (x + 2)(x + 3).
* Grouping: If an expression has four terms, we can try to group them into two pairs and factor each pair separately. Then we can factor out the common factor from the two pairs. For example, x^3 – x^2 – x + 1 = x^2(x – 1) – (x – 1) = (x – 1)(x^2 – 1) = (x – 1)(x + 1)(x – 1).
– **Linear equations**: A linear equation is an equation of the form ax + b = c, where a, b and c are constants and x is the unknown variable. To solve a linear equation, we can use the following steps:
* Isolate x by adding or subtracting the same term from both sides of the equation.
* Simplify both sides of the equation by combining like terms.
* Divide both sides of the equation by the coefficient of x.
* Check the solution by plugging it back into the original equation.
– **Quadratic equations**: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b and c are constants and x is the unknown variable. To solve a quadratic equation, we can use one of the following methods:
* Factoring: If the equation can be factored as (px + q)(rx + s) = 0, then we can use the zero product property to find the solutions as x = -q/p or x = -s/r.
* Completing the square: If the equation can be written as (x + h)^2 = k, then we can use the square root property to find the solutions as x = -h ± √k.
* Quadratic formula: If the equation is of the form ax^2 + bx + c = 0, then we can use the quadratic formula to find the solutions as x = (-b ± √(b^2 – 4ac)) / 2a.
– **Functions**: A function is a rule that assigns to each input value a unique output value. We can use the following notation and terminology to describe functions:
* f(x) = y means that the function f assigns the output value y to the input value x.
* The domain of a function is the set of all possible input values.
* The range of a function is the set of all possible output values.
* The graph of a function is the set of all ordered pairs (x, f(x)) in a coordinate plane.
* A function is linear if its graph is a straight line. A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept.
* A function is quadratic if its graph is a parabola. A quadratic function has the form f(x) = ax^2 + bx + c, where a, b and c are constants and a ≠ 0.
* A function is exponential if its graph has a constant ratio between consecutive y-values for equal changes in x. An exponential function has the form f(x) = ab^x, where a and b are constants and b > 0.
These are some of the basic algebra formulas that you can use to solve various problems. For more information and examples, you can visit this website
(1) Algebra Formulas | List of Algebraic Expressions in Maths – BYJU’S. https://byjus.com/algebra-formulas/.
(2) Algebra (all content) | Khan Academy. https://www.khanacademy.org/math/algebra-home.
(3) Algebra Formulas, All Algebraic Identities Formulas PDF – Adda247. https://www.adda247.com/school/algebra-formulas/.
(4) Algebra Formulas – Algebraic Formulas for Class 8, 9, 10, 11, 12. https://www.cuemath.com/algebra/algebraic-formulas/.