The standard form is referred to as the general method of representing any type of notation. A standard form formula represents the commonly accepted form of an equation, which is the standard form.
As an example – The standard form of a polynomial is to write the terms with the highest degree first (descending order of degree) and its coefficients should be expressed as integrals. By using the standard form formula, different types of notation can be represented in a general way.
Table of Contents
What Is the Standard Form Formula?
The degree of an equation determines the formula to represent the standard form. This article looks at simple linear equations and quadratic equations in their standard form.
Standard Form Formula
- A linear equation has a standard form, which is the basic form of an equation. A linear equation with two variables and more than two variables is presented below in its standard form. Here x, y, or \(x_1, x_2, x_3\),…. represent the variables and a, b, \(A_1, A_2, A_3, …….. A_n\) are referred to as the coefficients. Constants are numbers placed immediately before an equals sign. ax + by = c\(A_{1}x_{1} +A_{2} x_{2}+ A_{3} x_{3} + …….. + A_{n} x_{n}\) = D
- Standard quadratic equations are second-degree equations with variable coefficients and constant terms. This equation has a single variable x of degree 2. The standard form of a quadratic equation is the following.
ax2 + bx + c = 0
where a ≠ 0
Additionally, we have standard form formulas for equations of higher degrees. In coordinate geometry, we have standard forms for different geometric representations like a straight line, a circle, an ellipse, a hyperbola, and a parabola.
- Straight-line: Ax + By = C, where A is a positive integer, and B, and C are integers.
- Circle: \((x – h)^2 + (y – k)^2 = (r^2)\), where ( h, k ) is the center and r is the radius.
- Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} =1\)
- Hyperbola: \(\dfrac{(x-x_{0})^{2}}{a^{2}}-\dfrac{(y-y_{0})^{2}}{b^{2}}=1\), where \(x_{0}\), \(x_{0}\) are the center points, a = semi-major axis and b = semi-minor axis.
- Parabola: \((x – h)^2 = 4p(y – k)\)
To better understand the Standard Form Formula, let’s look at a few examples.
Examples Using Standard Form Formula
Example 1: Convert the following linear equation y = 3x + 4 into standard form.
Solution:
y = 3x + 4
y – 3x = 4
-3x + y = 4
3x – y = -4
Answer: Standard form of the given linear equation is 3x – y = -4
Example 2: Convert the following quadratic equation into standard form: \( \frac{2x^2}{3} = 3x – 7\)
Solution:
\( \dfrac{2x^2}{3} = 3x – 7\)
Multiplying by 3 on both sides,
2x2 = 9x – 21
Shifting R.H.S. terms to L.H.S.,
2x2 – 9x + 21 = 0
Answer: Standard form of the given quadratic equation is 2x2 – 9x + 63 = 0
Example 3: Convert the following quadratic equation into standard form: \( \frac{1}{x} + x = 1\)
Solution:
\( \dfrac{1}{x} + x = 1\)
Multiplying x on both sides,
1 + x2 = x
Shifting R.H.S. terms to L.H.S.,
x2 – x + 1 = 0
Answer: Standard form of the given quadratic equation is x2 – x + 1 = 0
FAQs
What Is Standard Form Formula?
The formula presenting the general representation for different types of notation is known as the standard form formula.
What Is the Standard Form Formula for Parabola?
The standard form formula of the equation of the parabola is this: (y – k)2 = 4p(x – h), where p≠ 0 only in case a parabola has a horizontal axis.
- The vertex of this parabola is at (h, k).
- The focus is at (h + p, k).
What Is the Standard Form for Slope Formula?
The standard form of slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line whereas the standard form of a linear equation is Ax + By = C.
How To Use Standard Form Formula?
Standard form formula is used depending on the type of equation, whether linear, quadratic, etc. All you have to do is rewrite the formulas given in standard form.
- Standard form of linear equation: ax + by = c
- Standard form of a quadratic equation is a second degree equation: ax2 + bx + c = 0