Standard form is the method of representing complex things in simpler form. We have methods to write different equations and numbers in standard form. Here we are going how to deal with writing the standard form of different equations and numbers. We will also learn how to write the conic sections in standard form.

The equations are named based on the degree of the equations.

The equations with degree 1 are known as linear equations.

The equations of degree 2 are known as quadratic equations.

The equations of degree 3 are known as cubic equations and so on.

The standard form of a linear equation is given by Ax+ By = C where A, B and C are constants and x and y are variables

Example: -

Find the equation of the line passing through (6, -2) and having a slope 3.

Solution: -

We know that if a line passes through a point (x

Given that the line passing through (6, -2). That is (x

Slope is 3. That is m = 3

Therefore the required line is

y - (-2) = 3 (x - 6)

y + 2 = 3x - 18

adding 18 both sides, we get

y + 20 = 3x

subtracting y on both sides, we get

The standard form of the quadratic equation is ax

Write the quadratic equation in standard form.

2x

We know that the standard form of the quadratic equation is given by ax

Given equation is 2x

To convert this to standard form, we take all terms to left.

Subtracting 12x and 3 on both sides, we get

A

If a > 0, the parabola opens upwards and if a < 0, the parabola opens downwards.

Example: -

Write the equation of the parabola whose focus is S(-4,5) and vertex (-4,1).

Solution: -

Distance between the focus and the vertex = 4

Distance between the vertex and the directrix = 4

Distance between the vertex and x axis = 1

Therefore distance between x axis and directrix =3

Since the directrix is parallel to x axis and at a distance of 3 units below the x axis, the equation to the directrix is y = -3

Let P(x,y) be any point on the parabola. Let PM be the perpendicular to the directrix.

Then SP

Since SP

(x + 4)

Simplifying we get

x

Cancelling common terms, we get

x

subtracting 6y and 9 on both sides, we get

x

Write all terms in x and constants on one side

16y = x

Dividing by 16 on both sides, we get

Write the equation of the circle in the standard form, whose center is (6, 9) and radius is 2.

Given center of the circle,(h, k) = (6, 9) and radius of the circle, r = 2.

We know that the standard form of a circle is (x - h)

Substituting for h, k and r, we get

(x - 6)

Simplifying we get,

(x - 6)

Write the number 47956.759 in the standard form.

Given number is 47956.759. we have to write this number in standard form.

In the standard form we write the number in the form x * 10

That is 47956.759 = 4.7956759 x 10

- First we have to take the common factors from the first two terms in the standard form f(x) = ax
^{2}+ bx + c - Now we complete the square of the first two terms by adding and subtracting square of the half of second term.
- Simplify and rearrange it in the form y = a(x - h)
^{2}+ k

Example: -

Convert the parabola f(x) = x^{2} - 6x + 6 in the standard form to vertex form and hence find the vertex.

Solution: -

Given standard form of parabola is f(x) = x^{2} - 6x + 6.

Comparing with the standard form of parabola, f(x) = ax^{2} + bx + c, we get

a = 1.

Since
a = 1, we complete the square of the first two terms in the quadratic
equation. To complete the square, we add and subtract the square of the
half of the second term.

So we get

f(x) = x^{2} - 6x + 9 - 9 + 6

Now we can combine the first three terms and last two term.

We get,**f(x) = (x - 3) ^{2} - 3, which is the required vertex form and the vertex is given by (3, -3).**

When we take all the variables to one side and keep the constants on the other side, we get the standard form.

**Example: -**

Convert the line y = 5x + 8 into standard form.**Solution: -**

Given
line is y = 5x + 8. To convert into standard form, we have to take all
variables to one side, keeping the constants on the other side.

For this lets subtract 5x from both sides,

y = 5x + 8__-5x -5x __

y - 5x = 8**So the required equation is -5x + y = 8**

The standard form of a quadratic function is given by

f(x) = a(x - h)

This form is also known as vertex form of a quadratic function

When we arrange the polynomial in the decreasing order of its degree, it is said to be in the standard form. In general, an nth degree polynomial P(x) = a