Standard Form of Quadratic Equation

The standard form of quadratic equation is ax² + bx + c = 0, where 'a' is the leading coefficient and it is a non-zero 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.

The standard form of quadratic equation with a variable x is of the form ax² + bx + c = 0, where a ≠ 0, and a, b, and c are real numbers. Here, b and c can be either zeros or non-zero numbers and

• 'a' is the coefficient of x²
• '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.

• 2x² - 7x + 8 = 0
• (-1/3) x² + 2x - 1 = 0
• √2 x² - 8 = 0
• -3x² + 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 ax² + bx + c = 0, where a ≠ 0.

Let us convert the standard form of a quadratic equation ax² + 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.

ax² + bx + c = a (x - h)² + k
ax² + bx + c = a (x² - 2xh + h²) + k
ax² + bx + c = ax² - 2ah x + (ah² + k)

Comparing the coefficients of x on both sides,
b = -2ah ⇒ h = -b/2a ... (1)

Comparing the 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)

Thus, we can use the formulas h = -b/2a and k = (4ac - b²) / (4a) to convert the standard to vertex form.

Example of Converting Standard Form to Vertex Form

Consider the quadratic equation 2x² - 4x + 3 = 0. Comparing this with ax² + 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 - b²) / (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 (x² - x - x + 1) + 1 = 0
2 (x² - 2x + 1) + 1 = 0
2x² - 4x + 2 + 1 = 0
2x² - 4x + 3 = 0 --------> Standard Form

Let us convert the standard form of a quadratic equation ax² + bx + c = 0 into the vertex form a (x - p)(x - q) = 0. Here, (p, 0) and (q, 0) are the x-intercepts of the quadratic function f(x) = ax² + 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 2x² - 7x + 5 = 0. By comparing this with ax² + bx + c = 0, we get a = 2. Now we will solve the quadratic equation by factorization.

2x² - 7x + 5 = 0
2x² - 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
2x² - 5x - 2x + 5 = 0
2x² - 7x + 5 = 0 --------> Standard Form

Important Notes on Standard Form of Quadratic Equation:

• A quadratic equation in standard form is ax² + bx + c = 0.
• A quadratic equation in vertex form is a (x - h)² + k = 0, where h = -b/2a and k = (4ac - b²) / (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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What is Quadratic Standard Form?

The standard form of a quadratic equation with variable x is expressed as ax² + bx + c = 0, where a, b, and c are constants such that 'a' is a non-zero 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 ax² + 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:

• √3x² - 7x + 2 = 0
• x² - 4x + (1/2) = 0
• 0.5x² - 18 = 0
• x² + 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) = ax² + bx + c, where a ≠ 0.

How to Convert a Quadratic Equation From Standard to Vertex Form?

The standard form of a quadratic equation is ax² + 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 - b²) / (4a)

Can 'a' be a Zero in Standard Form of Quadratic Equation?

The standard form of quadratic equation is ax² + 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 ax² + bx + c = 0 where 'a' is non-zero.

Can 'b' be a Zero in the Standard Form of a Quadratic Equation?

In the quadratic equation ax² + bx + c = 0, only 'a' (the coefficient of x²) shouldn't be a zero. So the value of 'b' can be 0.