Standard Form of the Quadratic Equation is ax² + bx + c = 0, where a, b, and c are constants and x is a variable. 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

Standard Form of Quadratic Equation

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.

Standard Form of Quadratic Equation

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

Learn More

• Quadratic Function
• Standard Equation of Parabola

Convert 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.

Convert 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

Convert 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.

Read More

• Quadratic Formula
• Roots of Quadratic Equations
• Relationship between Zeroes and Coefficients of a Polynomial

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

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

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

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

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

Standard Form of Quadratic Equation – FAQs

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.

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. 

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. 

What is Standard Form Formula for Polynomials?

Standard form formula for an n degree polynomial is:

a₁xⁿ + a₂x^n-1 + a₃x^n-2 +. . . + a_nx + c = 0 

Where, 

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

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

How do you Write a Quadratic Equation in Standard Form?

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

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.