Standard Form of Quadratic Equation
The standard form of quadratic equation is ax^{2} + bx + c = 0, where 'a' is the leading coefficient and it is a nonzero real number. This equation is called 'quadratic' as its degree is 2 because 'quad' means 'square'. Apart from the standard form of quadratic equation, a quadratic equation can be written in other forms.
 Vertex Form: a (x  h)^{2} + k = 0
 Intercept Form: a (x  p)(x  q) = 0
Let us learn more about the standard form of a quadratic equation and let us see how to convert one form of a quadratic equation into another.
What is the Standard Form of Quadratic Equation?
The standard form of quadratic equation with a variable x is of the form ax^{2} + bx + c = 0, where a ≠ 0, and a, b, and c are real numbers. Here, b and c can be either zeros or nonzero numbers and
 'a' is the coefficient of x^{2}
 'b' is the coefficient of x
 'c' is the constant
Examples of Standard Form of Quadratic Equation
Here are some examples of quadratic equations in standard form.
 2x^{2}  7x + 8 = 0
 (1/3) x^{2} + 2x  1 = 0
 √2 x^{2}  8 = 0
 3x^{2} + 8x = 0
General Form of Quadratic Equation
The standard form of a quadratic equation is also known as its general form. Thus, the general form of a quadratic equation is also of the form ax^{2} + bx + c = 0, where a ≠ 0.
Converting Standard Form of Quadratic Equation into Vertex Form
Let us convert the standard form of a quadratic equation ax^{2} + bx + c = 0 into the vertex form a (x  h)^{2} + k = 0 (where (h, k) is the vertex of the quadratic function f(x) = a (x  h)^{2} + k). Note that the value of 'a' is the same in both equations. Let us just set them equal to know the relation between the variables.
ax^{2} + bx + c = a (x  h)^{2} + k
ax^{2} + bx + c = a (x^{2}  2xh + h^{2}) + k
ax^{2} + bx + c = ax^{2}  2ah x + (ah^{2} + k)
Comparing the coefficients of x on both sides,
b = 2ah ⇒ h = b/2a ... (1)
Comparing the constants on both sides,
c = ah^{2} + k
c = a (b/2a)^{2} + k (From (1))
c = b^{2}/(4a) + k
k = c  (b^{2}/4a)
k = (4ac  b^{2}) / (4a)
Thus, we can use the formulas h = b/2a and k = (4ac  b^{2}) / (4a) to convert the standard to vertex form.
Example of Converting Standard Form to Vertex Form
Consider the quadratic equation 2x^{2}  4x + 3 = 0. Comparing this with ax^{2} + bx + c = 0, we get a = 2, b = 4, and c = 3. To convert it into the vertex form, let us find the values of h and k.
 h = b/2a = (4) / (2 · 2) = 1
 k = (4ac  b^{2}) / (4a) = (4 · 2 · 3  (4)^{2}) / (4 · 2) = (24  16) / 8 = 1
Substituting a = 2, h = 1, and k = 1 in the vertex form a (x  h)^{2} + k = 0, we get:
2 (x  1)^{2} + 1 = 0
Converting Vertex Form to Standard Form
The process of converting the vertex form of a quadratic equation into the standard form is pretty simple and it is done by simply evaluating (x  h)^{2} = (x  h) (x  h) and simplifying. Let us consider the above example 2 (x  1)^{2} + 1 = 0 and let us convert it back into standard form.
2 (x  1)^{2} + 1 = 0 > Vertex Form
2 (x  1) (x  1) + 1 = 0
2 (x^{2}  x  x + 1) + 1 = 0
2 (x^{2}  2x + 1) + 1 = 0
2x^{2 } 4x + 2 + 1 = 0
2x^{2}  4x + 3 = 0 > Standard Form
Converting Standard Form of Quadratic Equation into Intercept Form
Let us convert the standard form of a quadratic equation ax^{2} + bx + c = 0 into the vertex form a (x  p)(x  q) = 0. Here, (p, 0) and (q, 0) are the xintercepts of the quadratic function f(x) = ax^{2} + bx + c) and hence p and q are the roots of the quadratic equation. Thus, we just use any one of the solving quadratic equation techniques to find p and q.
Example to Convert Standard to Intercept Form
Consider the quadratic equation in standard form 2x^{2}  7x + 5 = 0. By comparing this with ax^{2} + bx + c = 0, we get a = 2. Now we will solve the quadratic equation by factorization.
2x^{2}  7x + 5 = 0
2x^{2}  2x  5x + 5 = 0
2x (x  1)  5 (x  1) = 0
(x  1) (2x  5) = 0
x  1 = 0; 2x  5 =0
x = 1; x = 5/2
Thus, p = 1 and q = 5/2
Thus, the intercept form is,
a (x  p)(x  q) = 0
2 (x  1) (x  5/2) = 0
2 (x  1) (2x  5)/2 = 0
(x  1) (2x  5) = 0
Converting Intercept Form to Standard Form
The process of converting the intercept form of a quadratic equation into standard form is really easy and it is done by simply multiplying the binomials (x  p) (x  q) and simplifying. Let us consider the above example (x  1) (2x  5) = 0 and let us convert it back into standard form.
(x  1) (2x  5) = 0 > Intercept Form
2x^{2}  5x  2x + 5 = 0
2x^{2}  7x + 5 = 0 > Standard Form
Important Notes on Standard Form of Quadratic Equation:
 A quadratic equation in standard form is ax^{2} + bx + c = 0.
 A quadratic equation in vertex form is a (x  h)^{2} + k = 0, where h = b/2a and k = (4ac  b^{2}) / (4a).
 A quadratic equation in intercept form is a (x  p)(x  q) + k = 0, where p and q are the roots of the quadratic equation.
☛Related Topics:
 Quadratic Expressions
 Multiplying Binomials Calculator
 Quadratic Equation Calculator
 Roots of Quadratic Equation Calculator
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Examples of Quadratic Equations in Standard Form

Example 1: Which of the following quadratic equations are in the standard form?
a) 2x^{2}  3x  5 = 0
b) 0x^{2}  3x + 5 = 0Solution:
We know that the standard form of a quadratic equation is ax^{2} + bx + c = 0, where 'a' is not equal to 0. Thus, in the given equations, only (a) is in the standard form
☛Note: Part b) has a = 0 and hence it becomes 3x + 5 = 0, which is a linear equation.
Answer: a) 2x^{2}  3x  5= 0 is in the standard form.

Example 2: Write the standard form of quadratic equation for the given expression: (x  7) ( x  8) = 0
Solution:
Let us convert the given equation into the standard form of quadratic equation (ax^{2} + bx + c = 0)
(x  7) ( x  8) = 0
x^{2}  7x  8x + 56 = 0
x^{2}  15x + 56 = 0Answer: The standard form of the given quadratic equation is x^{2}  15x + 56 = 0.

Example 3: Convert the following quadratic equation from standard to vertex form: 3x^{2}  18x + 1 = 0.
Solution:
Comparing the given equation with the standard form of quadratic equation ax^{2} + bx + c = 0, we get a = 3, b = 18, and c = 1. Now we will find h and k.
 h = b/2a = (18) / (2 · 3) = 3
 k = (4ac  b^{2}) / (4a) = (4 · 3 · 1  (18)^{2}) / (4 · 3) = (12  324) / 12 = 26
Substituting all these values in the vertex form a(x  h)^{2} + k = 0, we get:
3 (x  3)^{2}  26 = 0
Answer: The given quadratic equation in vertex form is 3 (x  3)^{2}  26 = 0.
FAQs on Standard Form of Quadratic Equation
What is Quadratic Standard Form?
The standard form of a quadratic equation with variable x is expressed as ax^{2} + bx + c = 0, where a, b, and c are constants such that 'a' is a nonzero number but the values of 'b' and 'c' can be zeros.
Can 'c ' be a Zero in the Standard Form of Quadratic Equation?
In the standard form of quadratic equation ax^{2} + bx + c = 0, only 'a' has a restriction that it should not be a zero. So the value of 'c' can be 0. Note, that the value of 'b' can also be a zero.
What is an Example of Quadratic Equation in Standard Form?
Some examples of quadratic equations in standard form are:
 √3x^{2}  7x + 2 = 0
 x^{2}  4x + (1/2) = 0
 0.5x^{2}  18 = 0
 x^{2} + x = 0
What is the Standard Form of Quadratic Function?
A quadratic function is a function that involves quadratic expression. i.e., its equation in standard form is f(x) = ax^{2} + bx + c, where a ≠ 0.
How to Convert a Quadratic Equation From Standard to Vertex Form?
The standard form of a quadratic equation is ax^{2} + bx + c = 0. To convert it into the vertex form a(x  h)^{2} + k = 0,
 The value of 'a' is obtained from the standard form.
 h = b/2a
 k = (4ac  b^{2}) / (4a)
Can 'a' be a Zero in Standard Form of Quadratic Equation?
The standard form of quadratic equation is ax^{2} + bx + c = 0. If a = 0, then the equation becomes bx + c = 0 which is not quadratic anymore (this is actually linear). Thus, the value of 'a' should NOT be a zero in a quadratic equation.
What is the Difference Between the General Form of Quadratic Equation and the Standard Form of Quadratic Equation?
The standard form of a quadratic equation is as same as its general form and is expressed as ax^{2} + bx + c = 0 where 'a' is nonzero.
Can 'b' be a Zero in the Standard Form of a Quadratic Equation?
In the quadratic equation ax^{2} + bx + c = 0, only 'a' (the coefficient of x^{2}) shouldn't be a zero. So the value of 'b' can be 0.
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