In algebra 2 we are going to study equations and inequalities more deeply.

Topics included in algebra 2 besides equations and inequalities are: polynomials and factorization, fractional expressions, power and root, graphs and functions, system of linear equations and matrices, co-ordinate geometry.

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Determinant Calculator 2x2 | eigenvalue calculator 2x2 |

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We must clear all the basic concepts in algebra 1 in order to make the study of algebra 2 easier i.e. real numbers, order of operations, algebraic expressions. Theses topics form a strong base for the actual study of algebra.

__Linear equations and systems of linear equations__:

Linear equations are equations of lines. 2x = 10, y = 5, 2x – 8y = 17 such type of equations are called as linear equations.

Both variables x and y possess the same degree and it is 1 always. Graph of linear equation has been always a straight line.

Sometimes there are two or more equations working together i.e. they possess same solution set of all the equations.

This is called as system of linear equations. We can solve a system of equations using various methods.

Some of the methods are graphing equations, substitution method, elimination method.

Solve the linear equations using substitution method

x + 2y = 10

3x + 4y = 20

We have x + 2y = 10 which implies x = 10 - 2y -----(i)

Substitute the value of x in equation 3x + 4y = 20

We get 3 (10 - 2y) + 4y = 20

30 - 6y + 4y = 20

30 - 2y = 20

2y = 30 - 20

2y = 10

And y = 5

Substitute this value of y in equation 1, we get

x = 10 - 2 (5) = 10 - 10 = 0

So x = 0 and y = 5

**Quadratic equations:**

A quadratic equation is equations of the type: y = ax^{2} + bx + c, here the degree of x is always 2 and a, b and c are real constants.

A graph of quadratic equations looks like a parabola.

It is always symmetric. Axis of symmetry is given by x = $\frac{-b}{2a}$ and solutions for this equation is given by the quadratic formula:

x = $\frac{-b\pm \sqrt{b^2-4ac{^{}}}}{2a}$. Vertex of a parabola or curve is ($\frac{-b}{2a}$, f($\frac{-b}{2a}$))

**Example: **Find axis of symmetry for the parabola y = x$^2$ + 2x - 10

Axis of symmetry is given by x = $\frac{-b}{2a}$

We have a = 1 and b = 2 so axis of symmetry is x = -1

__Polynomials:__

Polynomials are represented as: p (x) = a_{n} x^{n} + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + … + a_{1} x + a_{0}

This is the polynomial of degree n where a_{n}, a_{n-1}, … , a_{1}, a_{0} are real coefficients. a_{n} is called as a leading coefficient while a_{0} is a constant. Each polynomial has number of zeros or roots equal to its degree. i.e. polynomial of degree 4 has 4 roots.

Roots may be repeated. Similarly it has a number of factors equal to its degree.

We can draw graphs of polynomials by evaluating number of coordinate pairs lying on this polynomial.

If p (x) = (x – a) (x – b) (x – c) is a polynomial in a factored form then a, b and c are roots or zeros of a polynomial.

**Example**:

Simplify: (x^{4} - 1)

We can write this expression as (x^{4 }- 1^{4})

(x^{2 }+ 1^{2}) (x^{2} - 1^{2})

(x^{2} + 1) (x + 1) (x - 1)

__Matrices:__

Matrices form to represent a system of linear equations. If we have a number of equations with a number of variables then we represent this system in matrix form and can solve this system using various methods like Gauss-Jordan method and elimination method.

Linear equations are equations of lines. 2x = 10, y = 5, 2x – 8y = 17 such type of equations are called as linear equations.

Both variables x and y possess the same degree and it is 1 always. Graph of linear equation has been always a straight line.

Sometimes there are two or more equations working together i.e. they possess same solution set of all the equations.

This is called as system of linear equations. We can solve a system of equations using various methods.

Some of the methods are graphing equations, substitution method, elimination method.

Solve the linear equations using substitution method

x + 2y = 10

3x + 4y = 20

We have x + 2y = 10 which implies x = 10 - 2y -----(i)

Substitute the value of x in equation 3x + 4y = 20

We get 3 (10 - 2y) + 4y = 20

30 - 6y + 4y = 20

30 - 2y = 20

2y = 30 - 20

2y = 10

And y = 5

Substitute this value of y in equation 1, we get

x = 10 - 2 (5) = 10 - 10 = 0

So x = 0 and y = 5

A quadratic equation is equations of the type: y = ax

A graph of quadratic equations looks like a parabola.

It is always symmetric. Axis of symmetry is given by x = $\frac{-b}{2a}$ and solutions for this equation is given by the quadratic formula:

x = $\frac{-b\pm \sqrt{b^2-4ac{^{}}}}{2a}$. Vertex of a parabola or curve is ($\frac{-b}{2a}$, f($\frac{-b}{2a}$))

Axis of symmetry is given by x = $\frac{-b}{2a}$

We have a = 1 and b = 2 so axis of symmetry is x = -1

Polynomials are represented as: p (x) = a

This is the polynomial of degree n where a

Roots may be repeated. Similarly it has a number of factors equal to its degree.

We can draw graphs of polynomials by evaluating number of coordinate pairs lying on this polynomial.

If p (x) = (x – a) (x – b) (x – c) is a polynomial in a factored form then a, b and c are roots or zeros of a polynomial.

Simplify: (x

We can write this expression as (x

(x

(x

Matrices form to represent a system of linear equations. If we have a number of equations with a number of variables then we represent this system in matrix form and can solve this system using various methods like Gauss-Jordan method and elimination method.