Quadratic Sloving Equations

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A quadratic function is a second degree polynomial function of the formquadratic function where a,b and c are constants. The expression ax2 + bx + c is called quadratic expression. If the expression is set to equal to zero, then it is called as quadratic equation. The roots of the quadratic equation are the solutions of the quadratic function. The graph of a quadratic function is plotted by taking x values along the x axis and f(x) values along y axis. The graph of a quadratic function is a parabola whose major axis is parallel to y axis.

Quadratic equation forms - Explained

There are three forms of quadratic equations

General Form: The quadratic function of the formquadratic function where a, b and c are constants.
Factored Form: The quadratic function of the form f(x) = ( x - x1 ) ( x - x2 ) where x1 and x2 are the roots of the quadratic equation
Vertex Form: The quadratic function of the form f(x) = a ( x - h )2 + h where h and k are the coordinates of the vertex. This form is also named as standard form of parabola

General form of a quadratic function

For the quadratic ...
... function of the form f(x) = ax2 + b x + c where a, b and c are constants.

The x-intercepts are the solutions to the equation 0 = ax2 + b x + c given by x intercepts
The y-intercept is (0,c) obtained by replacing x by zero.
The vertex is (`(-b)/(2a)` , f ( `(-b)/(2a)` ) )
Parabola has no x-intercepts, if the discriminant is negative that is the quadratic function has imaginary roots.

Factored form of quadratic function

For the quadratic function of the form f(x) = a ( x -r1 )( x - r2 )

The x - intercepts are ( r1,0 ) and ( r2,0 )

The y - intercept is ( 0 , ar1r2)

The vertex is vertex of the quadratic function in factored form

Points to remember

The graph of a quadratic function is a parabola. Irrespective of the forms, the leading coefficient 'a' of the quadratic functions decides the nature of the graph.

If a 0, the parabola opens upward.
If a 0, the parabola opens downwards

x-intercepts: The x-intercepts are also called the roots of the function. They are specifically the zeroes of the function.
y-intercept: The y-intercept is an initial value or initial condition, especially the independent variable represents.
Vertex: The vertex represents the maximum (or minimum) value of the function

Algebra is a branch of mathematics. Algebra plays an important role in our day to day life. Learn algebra ii covers the four basic operations such as addition, subtraction, multiplication and division. The most important terms of practice algebra course are variables, constant, coefficients, exponents, terms and expressions. In Algebra, besides numerals we use symbols and alphabets in place of unknown numbers to make a statement. Hence, learn algebra ii may be regarded as an extension of Arithmetic.

Topics to learn in algebra ii

Learning equations in algebra ii:

Variables of the algebra is combined to form algebraic equation by the following symbols such as addition (+), subtraction (-), multiplication (*) ,division (/) and equal to (=). Those variables may or may not have powers and exponents. The followings are the examples of the equations,

1. x2+x+2=0

2. x2+x=0

3. x2+2=0

4. x2=0 and etc…,

Normally, Equations are used for balancing the both sides of the equal sign(=) and also used for finding the value of the variable.

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Learning inequations in algebra ii:

Variables of the algebra is combined to form algebraic inequality by the following symbols such as addition (+), subtraction (-), multiplication (*) ,division (/), equal to (=), greater than (), greater than or equal to (=), less than (), or less than or equal to (=). Those variables may or may not have powers and exponents. The followings are the examples of the inequalities,

1. x2+x+2 0

2. x2+x = 0

3. x2+2 0

4. x2 = 0 and etc…,

Normally, inequalities will show both sides of the values never equals.

Learning polynomials in algebra ii:

Two different algebraic terms are combined by the arithmetic operations such as addition, subtraction, multiplication and division to form the algebraic expression, such expression is called polynomials or multinomial.

Example: 2x3y + z

Learning linear equations in algebra ii:

Linear equations are the polynomial equations whose degree or power is one.

Example: 1. x + y = 3, 2. 5x + 8y - 9 = 0 and etc..,

Learning example problems in algebra ii:

Ex 1: Solve the equation by using the quadratic equation x2+3x+2=0

Sol: x2+3x+2=0

a = 1
b = 3
c = 2

Discriminant: b2-4ac = 32-4*1*2 = 1

Discriminant (1) is greater than zero. The equation has two solutions.

X=(-b±√b2-4ac)/2a

X=(-3±√32-4*1*2)/2*1

or

x1,2 = (-3 ± 1) / 2*1

or

x1 = -2 / 2 = -1

x2 = -4 / 2 = -2

Ex 2: Solve the equation by using the factorization method x2+3x+2=0

Sol: x2+3x+2=0

x2+2x+x+2=0

(x2+2x)+(x+2)=0

X(x+2)+(x+2)=0

(x+1)(x+2)=0

X+1=0 and x+2=0

X+1-1=0-1 and x+2-2=0-2

X=-1 and x=-2

Therefore, x=-1-2

Ex 3: 2x+y=8

-2x+y=0

Substitution method:

2x+y=8 ---------------------- equation 1
-2x+y=0---------------------- equation 2

if we add the equations 1 and 2, we will get

2y=8

2y/2=8/2 ( both sides are divided by 2 )

y=4

Substitue y=4 in the equation 1, so we will get

2x+4=8

2x+4-4=8-4 ( -4 is added on both sides)

2x=4

2x/2=4/2

X=2

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