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!
What is the Standard Form in Math? Definition, Examples
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 Definition
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.
Standard Form Math Examples & Explanation
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 108 . 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 is also the same, only power is minus. So, 8.35 x 108 & 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
Standard Form Equation in Math
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.
Standard Form Calculator in Math
We do a lot of calculations in calculators. When the number is very large or very small, a standard form is to be used. In the standard form 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 4.5×108. In the calculator, we simply follow the below steps,
- Make the Scientific mode on,
- Press 4.5, then press ‘Exp’ & then 8.
- It’s like 4.5 Exp 8
Calculating the standard form using conventional calculators could take lots of time. To ease up the calculations, an online standard form calculator can be very helpful and time-efficient.
It will be represented as 4.508 or 450000000 or 4.5 E8.
Standard Form of a Polynomial in Math
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 for Standard Form of a Polynomial in Math
- 5x2 + 9 + 21x6 + 3x5 + x3
Explanation for Standard Form of a Polynomial in Math
- Variable = x
- Highest variable = 6
- Power of variable goes from 6, 5, 3, 2 from the left side to the right side.
So, the standard form shall be, 21x6 + 3x5 + x3 + 5x2 + 9
Standard Form of a Circle in Math
The standard form of a circle is explained with an example.
Let’s us consider, an equation, x2 + y2 – 10x + 6y + 18 = 0. Now, we will write this equation in standard forms. We know that the standard form of a circle is written as, (x-a)2+(y-b)2=r2, where (a,b) is the center of the circle and c is the radius.
x2 + y2 – 10x + 6y + 18 = 0
x2 -10x + 25 + y2 + 6y + 9 -16 = 0
x2 -2.x.5 + 52 + y2 + 2.y.3 + 32 = 16
(x-5)2 + (y+3)2 = 42
Here, (5, -3) is the center and 4unit is the radius.
Standard Form for Quadratic Equation in Math
The standard form of quadratic equation is written as Ax2 + Bx + C = 0.
Example for Standard Form for Quadratic Equation in Math
- x(x−4) = 25
Explanation for Standard Form for Quadratic Equation in Math
If we simplify, x(x−4) = 25 x2 -4x = 25 x2 -4x – 25 =0,
In this equation,
- A = 1,
- B = −4,
- C = −25
This is in line with Ax2 + Bx + C = 0, i.e. standard form.
Standard Form for Linear Equations or a Line in Math
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 for Standard Form for Linear Equations or a Line in Math
- y = 5x −8
Explanation for Standard Form for Linear Equations or a Line in Math
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.
Standard Form of Parabola in Math
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 for Standard Form of Parabola in Math
Look at the standard form of a parabola, and find out the focus, vertex & directrix. (x-5)2=-12(y-2)
Explanation for Standard Form of Parabola in Math
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.
Standard Form of Hyperbola in Math
The standard form equation of a hyperbola is written as,
- (x-h)2/a2-(y-k)2/b2=1 for horizontal hyperbola
- (y-k)2/a2-(x-h)2/b2=1 for vertical hyperbola
The Centre of a hyperbola is at (h, k) in both horizontal & vertical hyperbolas.
It fulfills the equation, c2=a2+b2
Where c is equal to the distance between the center and focus point.
Example for Standard Form of Hyperbola in Math
(x-3)2/4-(y-5)2/9=1, find out the center and the distance from the center to focus.
Explanation for Standard Form of Hyperbola in Math
(x-3)2/4-(y-5)2/9=1 can be written as (x-3)2/22-(y-5)2/32=1.
Hence, from the equation, we can write,
- h = 3,
- k = 5,
- a =2,
- b = 3,
c = a2+b2 =22 + 32 =4 + 9 = 13.
Center (3,5) and distance is 13 units.
Standard Form of Slope in Math
The standard form of slope is described as y = mx + C.
Example for Standard Form of Slope in Math
- y = 5x −8
Explanation for Standard Form of Slope in Math
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.
Standard Form of Decimal Numbers
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
Standard Form Examples & Calculation
Standard Form 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 105 km/s or 3 x 108 m/s.
Standard Form Example-2
How to write the standard form of the number 711000000.
It is written as 7.11 x 108
Standard Form 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.
Standard Form 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 104
Standard Form Example-5
Write a standard form of a quadric equation, 7y = – 2x2 +1
Here, the standard form of the above quadric equation will be, 2x2 + 7y – 1 = 0.
What is Standard Form?
Solved Problems on Standard Form
Anna showed her teacher the notes she was taking on polynomials. In returning her notes, the teacher wrote “Write the polynomial 2×2 – 20x + 32 – 8×2 + 4×5 – 6×4 + 7×2 in standard form.” What is the appropriate format Anna should have used?
Two rules must be followed in order for a polynomial to be written in standard form. The terms should be written in descending order of their power.
There must be a difference between all terms.
We will arrange them in descending order first.
2×2 – 20x + 32 – 8×2 + 4×5 – 6×4 + 7×2
= 4×5 – 6×4 + 2×2 – 8×2 + 7×2 – 20x +32
We get the following when we add like terms 4×5 – 6×4 +x2 – 20x + 16.
Answer: 4×5 – 6×4 +x2 – 20x + 16 is the appropriate format Anna should use.
Is it possible to standard form y = 8/10 x + 3/5?
We need to multiply each fraction by 10 to clear the fractions because fractions aren’t allowed in standard form.
10 y = 8x + 6
10y – 6 = 8x +6 -6
10y – 6 = 8x
8x = 10y – 6
8x -10y = – 6
Answer: Standard form = 8x -10y = – 6
Please rewrite the following quadratic equation into standard form: (1/x) – x = 2?
(1/x) – x = 2
Multiplying both sides by x,
1 – x2 = 2x
The R.H.S. terms are changed to L.H.S.
x2 + 2x + 1 = 0
(x+1) ² = 0
Answer: (x+1) ² = 0 is the standard form of this equation.
Write the below polynomial into standard form: 2y² – 20y + 32 – 4y² +4 y⁵ – 6y⁴ + 6y²
2y² – 20y + 32 – 4y² +4 y⁵ – 6y⁴ + 6y²
= 4 y⁵ – 6y⁴ + 2y² + 6y² – 4y² – 20y + 32
= 4 y⁵ – 6y⁴ + 8y² – 4y² – 20y + 32
= 4 y⁵ – 6y⁴ + 8y² – 4y² – 20y + 32
Answer: 4 y⁵ – 6y⁴ + 8y² – 4y² – 20y + 32 is the standard form of this polynomial.
Change – 4/7 x – 9/4 y = 8 into standard form
The first step is to remove fractions by multiplying each term by the LCM. Here, 7 and 4 have a LCM of 28.
Why did you not write standard form in fraction?
You cannot write any equation into fraction because you have to maintain ‘Ax +By = C’ must be the form. The three numbers A, B, and C must be integers & not in fractions and A cannot be negative. All three numbers should have only one common factor.
You can solve for m (c) if you encounter an equation in slope-intercept form but need it in standard form. Fractions cannot be expressed in standard form. Multiply each side to get rid of them if they are fractions.
Can you write 9874 in standard form?
9874 can be written as 9.874 x 1000
As a result, the standard form of 9874 is 9.874 x 10³.
How you put a line in a standard form?
A conversion from one form of an equation to another should always be possible. We should be able to express a line in slope-intercept form and vice versa. As a result, Ax + By=C is called the standard form of the line.
Can you prove that a quadratic equation is in standard form?
The equation of the second degree is a quadratic equation. It contains at least one squared term. A, b, c are numerical coefficients in the quadratic equation of ax² + bx + c = 0 to solve any equations.
Can you please describe Standard Form of Fraction?
We need to ensure that the numerator and denominator of fractions are co-prime numbers in the standard form. The standard form of a fraction is, therefore, also known as the simplest form of a fraction, since they have no common factor other than 1. Examples include 14/22 and 13/6. Because 13 and 6 are co-primes, 14/22 = 7/22 and 13/6 is already in its simplest form.
FAQs for Standard Form in Math
Mathematically-speaking, a number is represented as a standard form. The simplest form of decimal numbers is the standard form. It may be read and written easily. 0.002 is represented in decimal form as 2 × 10-3. There are times when reading or writing very large or very small numbers is very difficult. For this reason, we write them in standard form.
Standard form is defined as the process of writing a very large expanded form or smallest form of a number into standard formats.
The standard form of 845000000 is written as 8.45 x 108
There are various ways to write standard form in math. Let us take an example for a quadratic equation.
Example for Standard Form for Quadratic Equation in Math
x(x−3) = 25
Explanation for Standard Form for Quadratic Equation in Math
If we simplify,
x(x−3) = 25
or, x2 -3x = 25
or, x2 -3x – 25 =0,
In this equation,
A = 1,
B = −3,
C = −25
This is in line with Ax2 + Bx + C = 0, i.e. standard form.
Let us take an example of standard form for linear equation.
Linear Equations y = 3x −7
Explanation for Standard Form for Linear Equations or a Line
If we simplify,
y = 3x −7
or, -3x + y = -3 x + 3x – 7
or, -3x + y = -7
or, -3x +y + 7 = 0
or, 3x – y -7 = 0
3x – y -7 = 0, it is the standard form of Ax + By + C = 0
In this equation,
A = 3,
B = −1,
C = −7
This is in line with Ax + By + C = 0, i.e. standard form.
Linear equations in two variables are formulated as: Ax + By = C. As an example, 7x+8y=20 is a linear equation in standard form. The intercepts x and y can be easily determined if an equation is written in this form. You can also use this form to solve systems of two linear equations.
You should write the standard form of a number as follows:
Write the first digit from the given number.
After the first number, add the decimal point.
The number of digits after the first number has been multiplied by ten. You can write it in the power of 10.
Standard form is ax + by = c, where ‘a’ is a positive integer, ‘b’ is an integer, and ‘c’ is an integer.
Let the equation be
4(4x -12y +2) = 0
16x – 48y +8 = 0
16x – 48y = -8
16x/8 – 48y/8 = -8/8
2x -6y = -1
If a number in the Standard Form can be expressed as a product of a number between 1 and 10 and ten, then it can be said to be in the Standard Form. A decimal point has been moved 11 places clockwise, making the power positive and as a result, the decimal is positive. Because the decimal is moved 6 places right, the power is negative.
Numbers are normally written in standard notation. The 2.4 million can be denoted as 2400000 since 1 million contains six zeros. Thus, 2.4 million is written as 2400000.
It is a very convenient way to write down very large or very small numbers like 10³ = 1000, so 4 × 10³ = 4000.
So you can write 4000 = 4 × 10³ in standard form.
Standard form =6.543 x 1000
= 6.543 x 10 ³
Expanded form = 6 x 1000 + 5 x 100 + 4 x 10 + 3.
A number 450 in standard form would be written as 450, but we can rewrite this number as 4.50 * 102, which is a form of scientific notation. Therefore, a standard form is considered as the format most commonly used.
The standard form of a number is a decimal number with a power of 10 multiplied by it.
Numbers can be written in standard form to make them easier to read. In the standard form, numbers are written as follows:
Standard form = 777
Expanded form = 700 + 70 + 7
Written form = Seven hundred seventy seven.
1. Ax+By=C must be the form.
2. All three variables (A, B, C) must be integers.
3. It is not possible for A to be negative.
4. Each of the three variables (A, B, C) should have one common factor.
Linear equations in standard form look like this ax + by = c. A, B, and C are all real numbers & A and B are not both zero. However, c can be zero if it wants, since it is the favorite child, so it is given special advantages.
A straight line can be represented mathematically in two-point form. A line’s equation describes every point on the line, i.e., it is satisfied by every point on the line. A line’s equation can be found by using its two-point form given two points on it.
It is possible to write numbers in general form. Therefore, a two digit number ab can be written as ab = 10a + b.
It is always divisible by 11 the sum of a 2-digit number and the number obtained by reversing its digits.
The standard form of a quadratic equation is ax2 + bx + c = 0 where a, b, and c are constants so that a and b can be nonzero values but c can be zero.
Conclusion
Hence, we have a basic idea about standard form in math with so many examples & exercises. You can refer to our most interesting articles
