Standard Form of Quadratic Equation

The 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, such as,

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

Key Characteristics:

• Degree 2: The highest power of x is 2, which makes it a quadratic equation.
• Parabola: The graph of a quadratic equation is a parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.
• Vertex: The vertex of the parabola is located at, and it represents the minimum or maximum point of the graph, depending on the sign of a.
• Axis of Symmetry: The axis of symmetry is the vertical line x = −b/2a​, passing through the vertex.

Examples of Standard Forms of Quadratic Equations

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

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.

Steps to Convert 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: 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.

Solution:

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: 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

Solved Examples of Standard Forms of Quadratic Equations

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

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

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

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

Question 4. Convert the following quadratic equation from standard to Intercept form: 11x² + 8x + 3 = 0.