Standard Form Formula

Standard form is a common way of representing any notation/equation. The Standard Form Formula represents the general acceptance form of an equation. For a polynomial, the standard form can be represented as writing the terms with higher-order first to the lower order, and the coefficients of each term should be in integral form. The formula to represent an equation in standard form depends on the degree of an equation.

Standard Form Formula for Linear Equations

The standard form of a linear equation with two variables x and y is ax + by = c where a, b, and c are integers. Let’s consider a linear equation which consists of more than two variables i.e., x₁, x₂, x₃, ….,x_n and A₁, A₂, A₃,…., A_n are coefficients of each term then standard form formula of the equation is given by,

(A₁)x₁ + (A₂)x₂ + (A₃)x₃ + …. + (A_n)x_n = c

Standard Form Formula for Quadratic Equations

The standard form of the second-degree quadratic equation for a single variable x is ax² + bx + c = 0 where a, b, c are integers and a ≠ 0. For n degree equations the standard form is (A₁)xⁿ + (A₂)x^n-1 + (A₃)x^n-2 + …. + (A_n)x + c = 0 where A₁, A₂, A₃, … A_n are coefficients, n is the degree of the equation, x is a single variable and c is the constant numeric term. Many geometrical figures have standard form formulas which are listed below,

• The standard form formula for a circle is (x-h)² + (y-k)² = r² where (h, k) is the center of a circle and r is the radius.
• The standard form formula for an ellipse is given by( x²/a²) + (y²/b²) = 1 where a, b are lengths of semi-major and semi- minor axis. 
• The standard form formula for a parabola is (x-h)² = 4p(y-k).
• The standard form formula for a hyperbola is ((x – x₀)²/a²) – ((y – y₀)²/b²) = 1 where (x₀, y₀) are center point and a, b are lengths of semi major and semi minor axis. 

Sample Questions

Question 1: Convert the given linear/straight line equation y – 4 = -2x in standard form.

Solution:

Given linear equation

y – 4 = -2x

The standard form of linear equation is ax + by = c

2x + y – 4 = 0

2x + y = 4

So the standard form of given equation is 2x + y = 4.

Question 2: Convert the given straight line equation 7y = -2x + 3 in standard form.

Solution:

Given linear equation

7y = -2x + 3

The standard form of linear equation is ax + by = c

7y + 2x = 3

2x + 7y = 3

So the standard form of given equation is 2x + 7y = 3.

Question 3: 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.

Question 4: 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

Question 5: 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

Question 6: 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.

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