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Standard Form of Quadratic Equation

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Standard Form of the Quadratic Equation is ax2 + bx + c = 0, where a, b, and c are constants. Standard Form is a common way of representing any notation or equation, Quadratic equations can also be represented in other forms as,

  • Vertex Form: a(x – h)2 + k = 0
  • Intercept Form: a(x – p)(x – q) = 0

In this article we will learn about the standard form of the quadratic equation, changing it into the standard form of the quadratic equation and others in detail.

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Standard Form of Quadratic Equation

What is Standard Form of a Quadratic Equation?

Quadratic Equations are second-degree equations in a single variable and the standard form of Quadratic Equations is given as follows:

ax2 + bx + c = 0 

Where, 

  • a, b, and c are integers
  • a ≠ 0
  • ‘a’ is the coefficient of x2
  • ‘b’ is the coefficient of x
  • ‘c’ is the constant

Examples of Standard Form of a Quadratic Equation

Various examples of the quadratic equation in standard form are,

  • 11x2 – 13x + 18 = 0
  • (-14/3)x2 + 2/3x – 1/4 = 0
  • (-√12)x2 – 8x = 0
  • -3x2 + 9 = 0

General Form of Quadratic Equation

The general form of the quadratic equation is similarto the standard form of the quadratic equation. And the general form of the quadratic equation is, ax2 + bx + c = 0 where a, b and c are Real Numbers and a ≠ 0.

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Converting Quadratic Equations to Standard Form

Converting Quadratic Equations to Standard Form 

Step 1: Rearrange the equation so that the terms are in order of decreasing degree (from highest to lowest).

Step 2: Combine any like terms i.e., add and subtract like terms.

Step 3: Make sure that the coefficient ‘a’ of the x2 term is positive. If it’s negative, multiply the entire equation by -1.

Step 4: If there is any missing term i.e., term with x, add 0.x for that.

Example of Converting Quadratic Equations to Standard Form

Let’s understand the concept of Converting Quadratic Equations to Standard Form using the following example:

Example: Convert the following linear equation into Standard Form: 2x2 – 5x = 2x – 3 

Step 1: Rearrange the equation.

2x2 – 5x – 2x + 3 = 0

Step 2: Combine any like terms.

2x2 – 7x + 3 = 0

Step 3:  Coefficient of leading term is already positive, thus no need to multiply with -1.

Step 4: There are no missing terms of s.

Thus, 2x2 – 7x + 3 = 0 is the standard form of the given equation.

Converting Standard Form of Quadratic Equation into Vertex Form

We know that the standard form of a quadratic equation is ax2 + bx + c = 0 and the vertex form is a(x – h)2 + k = 0 (where (h, k) is the vertex of the quadratic function. 

Now we can easily convert the standard form into vertex form by comparing these two equations as,

ax2 + bx + c = a (x – h)2 + k

⇒ ax2 + bx + c = a (x2 – 2xh + h2) + k

⇒ ax2 + bx + c = ax2 – 2ahx + (ah2 + k)

Comparing coefficients of x on both sides,

b = -2ah

⇒ h = -b/2a … (1)

Comparing constants on both sides,

c = ah2 + k

⇒ c = a (-b/2a)2 + k (From (1))

⇒ c = b2/(4a) + k

⇒ k = c – (b2/4a)

⇒ k = (4ac – b2) / (4a)

Now the formulas h = -b/2a and k = (4ac – b2) /(4a) are used to convert the standard to vertex form.

Example of Converting Standard Form to Vertex Form

Consider the quadratic equation 3x2 – 6x + 4 = 0. Comparing this with ax2 + bx + c = 0, we get a = 3, b = -6, and c = 4. Now for vertex form, we found h and k

h = -b/2a 

⇒ h = -(-6) / (2.3) = 1

⇒ k = (4ac – b2) / (4a) 

⇒ k = (4.3.4 – (-6)2) / (4.3) 

⇒ k = (48 – 36) / 12 = 1

Substituting a = 3, h = 1, and k = 1, the vertex form a(x – h)2 + k = 0 is,

3(x – 1)2 + 1 = 0

Converting Vertex Form to Standard Form

We can easily convert the vertex form of a quadratic equation into the standard form by simply solving (x – h)2 = (x – h) (x – h) and simplifying. 

Let us consider the above example 2(x – 1)2 + 1 = 0 and convert it back into standard form.

3(x – 1)2 + 1 = 0                       (Vertex Form)

⇒ 3(x2 – x – x + 1) + 1 = 0

⇒ 3(x2 – 2x + 1) + 1 = 0

⇒ 3x2 – 6x + 3 + 1 = 0

⇒ 3x2 – 6x + 4 = 0…(i)                (Standard Form)

Equation (i) is the required standard form of the quadratic form.

Converting Standard Form of Quadratic Equation into Intercept Form

We know that the standard form of a quadratic equation is ax2 + bx + c = 0 and the vertex form is a(x – p)(x – q) = 0 where (p, 0) and (q, 0) are the x-intercept and y-intercept respectively.

Now we can easily convert the standard form into intercept form by solving quadratic equations as p and q are the roots of the quadratic equation.

Example of Converting Standard Form to Intercept Form

Consider the quadratic equation 3x2 – 8x + 4 = 0. Comparing this with ax2 + bx + c = 0, we get a = 3, b = -8, and c = 4. Now finding the roots of the quadratic equation as

3x2 – 8x + 4 = 0

⇒ 3x2 – (6+2)x + 4 = 0

⇒ 3x2 – 6x – 2x + 4 = 0

⇒ 3x(x – 2) -2(x – 2) =  0

⇒ (3x -2)(x – 2) = 0

⇒ (3x -2) = 0 and (x – 2) = 0

⇒ x = 2/3 and x = 2

Thus, the intercept form of the quadratic equation is,

a(x – p)(x – q) = 0

⇒ 3(x – 2/3)(x – 2) = 0

⇒ (3x -2)(x – 2) = 0

Converting Intercept Form to Standard Form

We can easily convert the vertex form of a quadratic equation into the standard form by simply solving (x – p)(x – q)  = 0 and simplifying. 

Let us consider the above example (3x -2)(x – 2) = 0 and convert it back into standard form.

(3x -2)(x – 2) = 0                       (Intercept Form)

⇒ 3x2 – 6x – 2x + 4 = 0

⇒ 3x2 – 8x + 4 = 0…(i)                  (Standard Form)

Equation (i) is the required standard form of the quadratic form.

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Examples of Quadratic Equations in Standard Form

Example 1: Convert the given quadratic equation 2x – 9 = 7x2 in standard form.

Solution:

Given quadratic equation,

2x – 9 = 7x2

The standard form of quadratic equation is ax2 + bx + c = 0

⇒ 2x = 7x2 + 9

⇒ 7x2 – 2x + 9 = 0

So the standard form of given equation is 7x2 – 2x + 9 = 0.

Example 2: Convert the given quadratic equation (2x/7)-1 = 2x2 in standard form.

Solution:

Given equation,

(2x/7) – 1 = 2x2 

⇒ (2x-7(1))/7 = 2x2

⇒ (2x-7)/7 = 2x2

⇒ 2x – 7 = 7(2x2)

⇒ 2x – 7 = 14x2

⇒ 14x2 – 2x + 7 = 0

So the standard form of given equation is 14x2 – 2x + 7 = 0

Example 3: Convert the given equation (2x3/x) + 4 = 2x in standard form.

Solution:

Given equation,

(2x3/x) + 4 = 2x

One of the x in x3 is cancelled by the x in denominator to form x2

⇒ 2x2 + 4 = 2x

⇒ 2x2 – 2x + 4 = 0

The above equation is further simplified to give x2 – x + 2 = 0

So the standard form of given equation is x2 – x + 2 = 0

Example 4: Convert the given quadratic equation into standard form (3/x) – 2x = 5.

Solution:

Given equation: (3/x) – 2x = 5

⇒ (3-2x(x))/x = 5

⇒ (3-2x2)/x = 5

⇒ 3-2x2 = 5x

⇒ 2x2 + 5x – 3 = 0

So the standard form of given quadratic equation is 2x2 + 5x – 3 = 0.

Practice Questions on Standard Form of Quadratic Equation

Q1. Convert the following quadratic equation from standard to vertex form: x2 – 4x + 1 = 0.

Q2. Convert the following quadratic equation from standard to intercept form: 2x2 + 9x + 24 = 0.

Q3. Convert the following quadratic equation from standard to vertex form: -4x2 – 12x + 16 = 0.

Q4. Convert the following quadratic equation from standard to Intercept form: 11x2 + 8x + * = 0.

Standard Form of Quadratic Equation-FAQs

1. What is Standard Form Formula?

Standard Form Formula is a common way of representing any notation or equation, as the Standard Form is accepted by a large group of people as Standard.

2. What is Standard Form Formula for Linear Equations?

The standard form of a linear equation with two variables x and y is given as follows:

ax + by = c 

Where a, b, and c are integers. 

3. What is the Standard Form of Quadratic Equation?

The standard form of quadratic equation is given as follows:

ax2 + bx + c = 0 

Where, 

  • a, b, and c are integers and 
  • a ≠ 0. 

4.What is Standard Form Formula for Polynomials?

Standard form formula for an n degree polynomial is:

a1xn + a2xn-1 + a3xn-2 +. . . + anx + c = 0 

Where, 

  • a1, a2, a3, … an are coefficients
  • n is the degree of the equation
  • x is a dependent variable
  • c is the constant numeric term

5. What are Examples of Quadratic Equations in Standard Form?

Various examples of quadratic equations in standard form are:

  • 3x2 – 4x + 2 = 0
  • x2 – 11x + (11/2) = 0
  • -x2 + 11 = 0, etc

6. How do you Write a Quadratic Equation in Standard Form?

A quadratic equation in standard form is written as, ax2 + bx + c = 0.

7. What is the Standard Form of a Quadratic Equation with Examples?

Standard Form of the quadratic equation is ax2 + bx + c = 0. And some of the examples of the quadratic equations are,

  • 2x2 + 5x – 11 = 0
  • 3x2 + 11x – 6 = 0, etc.


Last Updated : 29 Feb, 2024
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