Let’s explore Standard Form in Math!

We will learn, what is the standard form in math or equation in terms of various aspects, like circles, polynomials, quadric equations, lines, etc. A detailed understanding of standard forms is explained here. Let’s explore the standard form!

Let’s discuss, what is the standard form in math along with examples. As the name suggests, the standard form means how to write numerals in standard forms.

Standard form is defined as the process of writing a very large expanded form or smallest form of a number into standard formats.

It is not only applicable for numbers only, it is applicable for various other things as well, like other equations, lines, circles, etc.

Let’s try to understand the standard form with numbers. We get different kinds of numbers in our calculations, for example, 835000000 or 0.0000235.

Now, it is really difficult to write or use in the calculation and the philosophy of standard form comes into the picture!

So, how to write these numbers in standard forms?

It’s very simple! It should follow,

- Short format
- Easy to write & ready
- Standard for all

The standard form of 835000000 is written as 8.35 x 10^{8 }. In this number, we considered one number between 1 to 10 multiplied by the power of 10. Here, 8.35 means it belongs between 1 to 10 and it is multiplied by 10 to the power 8.

The standard form of 0.0000235 is written as 2.35 x 10^{-5}. It also the same, only power is minus.

So, 8.35 x 10^{8 }& 2.35 x 10^{-5 }are standard forms of numbers.

We will explain here, the followings

- Standard form equation
- Standard form calculator
- Standard form of polynomials
- Standard form of a circle
- Standard form in quadratic equations
- Standard form for linear equations
- Standard form equation of a line
- Standard form for parabola
- Standard form for hyperbola
- Standard form of slope
- Changing to slope-intercept form
- Standard form scientific notation
- Standard form graphing
- Standard form hyperbola
- Standard Form of Decimal Numbers
- Standard forms numbers

The standard form in an equation is illustrated with a simple example, y = -9x + 2. Look at the above equation, it’s not in a standard form, so, what to do?

- We need to move the x-term to the left side of the equation.
- So, add +9x to both sides.
- It becomes, 9x + y= – 9x + 2 + 9x or 9x + y = 2.
- Hence, 9x + y = 2 is the standard form.

We do a lot of calculations in calculators. When the number is very large or very small, a standard form to be used.

In the calculator, there is one button, ‘Exp’ which is used to express the power of 10.

Say, if you want to write, 450000000, then how to write it in standard form in the calculator!

We have learned in standard form in numbers that it can be written as 4.5×10^{8}.

In the calculator, we simply follow the below steps,

- Make the Scientific mode on,
- Press 4.5, then press ‘Exp’ & then 8.
- It like 4.5 Exp 8

It will be represented as 4.5^{08} or 450000000 or 4.5 E8

The standard form for polynomials is explained with examples. In case of a polynomial, the rule is very simple, only the power of variables shall be highest to lowest from the right side.

Example, 5**x ^{2}** + 9 + 21

Explanation

- Variable = x
- Highest variable = 6
- Power of variable goes from 6, 5, 3, 2 from the left side to right side.

So, the standard form shall be, 21**x ^{6 }**+ 3

The standard form of a circle is explained with an example.

Let’s us consider, an equation, x^{2} + y^{2} – 10x + 6y + 18 = 0

Now, we will write this equation in standard forms.

We know that standard form of a circle is written as, (x-a)^{2}+(y-b)^{2}=r^{2},

where (a,b) is the center of the circle and c is the radius.

x^{2} + y^{2} – 10x + 6y + 18 = 0

x^{2} -10x + 25 + y^{2} + 6y + 9 -16 = 0

x^{2} -2.x.5 + 5^{2} + y^{2} + 2.y.3 + 3^{2} = 16

(x-5)^{2} + (y+3)^{2} = 4^{2}

Here, (5,-3) is the center and 4unit is the radius.

The standard form of quadratic equation is written as Ax^{2} + Bx + C = 0.

Example x(x−4) = 25

Explanation

If we simplify, x(x−4) = 25 x^{2} -4x = 25 x^{2} -4x – 25 =0,

In this equation,

A = 1,

B = −4,

C = −25

This is in line with Ax^{2} + Bx + C = 0, i.e. standard form.

The standard form of linear equation is written as Ax + By = C. This is the same as the standard form of a line as well.

Example y = 5x −8

Explanation

If we simplify,

-5x + y = 5x −8 -5x

-5x + y = −8

-5x + y + 8 = 0

5x – y – 8 = 0, it is the standard form of Y = 5x −8

In this equation,

A = 5,

B = −1,

C = −8

This is in line with Ax + By + C = 0, i.e. standard form.

The standard form of parabola is written as, (x – h)^{2} = 4p (y – k), Where,

- (h, k + p) is the focus
- (h, k) is vertex
- y = k – p is the directrix

Example

Look at the standard form of a parabola, and find out the focus, vertex & directrix. (x-5)^{2}=-12(y-2)

Explanation

If we compare, (x-5)^{2}=-12(y-2) with the standard equation, (x – h)^{2} = 4p (y – k)

So, we get,

h = 5

k = 2

4p = -12

p = -3

Hence, we get,

- (5, -1) is the focus
- (5, 2) is vertex
- y = 2 – (-3) or y=5 is directrix

You can check **standard form questions & answers** on the internet.

The standard form equation of a hyperbola is written as,

- (x-h)
^{2}/a^{2}-(y-k)^{2}/b^{2}=1 for horizontal hyperbola - (y-k)
^{2}/a^{2}-(x-h)^{2}/b^{2}=1 for vertical hyperbola

The Centre of a hyperbola is at (h, k) in both horizontal & vertical hyperbolas.

It fulfills the equation, c^{2}=a^{2}+b^{2}

Where c is equal to the distance between the center and focus point.

Example

(x-3)^{2}/4-(y-5)^{2}/9=1, find out the center and the distance from the center to focus.

Explanation

(x-3)^{2}/4-(y-5)^{2}/9=1 can be written as (x-3)^{2}/2^{2}-(y-5)^{2}/3^{2}=1.

Hence, from the equation, we can write,

h = 3,

k = 5,

a =2,

b = 3,

c = a^{2}+b^{2 }=2^{2} + 3^{2 }=4 + 9 = 13.

Center (3,5) and distance is 13 units.

The standard form of slope is described as y = mx + C.

Example y = 5x −8

Explanation

If we simplify,

-5x + y = 5x −8 -5x

-5x + y = −8

-5x + y + 8 = 0

5x – y – 8 = 0, it is the standard form of Y = 5x −8

In this equation,

A = 5,

B = −1,

C = −8

This is in line with Ax + By + C = 0, i.e. standard form.

The standard form of decimals is explained here.

It is difficult to write or read the numbers like 0.000000572 or 0.0000799 or .000000895 etc.

To study these kinds of numbers, the standard form of decimals is required.

Any number in decimal form can be written as between one (1) to ten (10) multiplied by the power of ten (10).

For example,

0.000000572 can be written as 5.72 x 10^{-7}

0.0000799 can be written as 7.99 x 10^{-5}

0.000000895 can be written as 8.95 x 10^{-7}

**Example-1**

The **speed of light** is 3,00,000km/s or 30,00,00,000m/s.

How do we write this number in standard form?

It is wriiten as 3 x 10^{5} km/s or 3 x 10^{8 }m/s.

**Example-2**

How to write the standard form of the number 711000000.

It is written as 7.11 x 10^{8 }

**Example-3**

Write a standard form of a line, 7x = 2y + 1

Here, the standard form of the above line will be, 7x – 2y – 1 = 0.

**Example-4**

Write a standard form of a decimal number, 0.00012

Here, the standard form of the above decimal number will be, 1.2 x 10^{4}

**Example-5**

Write a standard form of a quadric equation, 7y = – 2x^{2} +1

Here, the standard form of the above quadric equation will be, 2x^{2 }+ 7y – 1 = 0.

**Conclusion**

Hence, we have a basic idea about standard form in math with so many examples & exercises.

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