Question

What is the difference between the standard form, vertex form, factored form?

Answer 1

We need to find the difference between the standard form, vertex form, factored form. The quadratic equation is an equation containing a single variable of degree 2. We find the standard form, vertex form, and factored form of the quadratic equation to get the desired result.

Complete step by step solution:
We are asked to find the difference between the standard form, vertex form, and factored form. We will be solving the given question using the concept of quadratic equations.
A quadratic equation is a polynomial equation of degree two which means the highest exponent in the polynomial is 2.
A coefficient is a numeric value that is used to multiply a variable.
For example:
6 is the numerical coefficient of the term 6z.
The standard form of the quadratic equation is given as follows,
$\Rightarrow y=a{{x}^{2}}+bx+c$
Here,
a is the numerical coefficient of the term ${{x}^{2}}$
b is the numerical coefficient of the term $x$
c is the constant term
The value of the variable a should not be equal to zero.
For example: ${{x}^{2}}+6x+9$
The vertex form of a quadratic equation is given as follows,
$\Rightarrow y=m{{\left( x-a \right)}^{2}}+b$
Here,
$\left( a,b \right)$ represent the vertex of the parabola
m is the leading coefficient
If the value of m is positive the graph of the parabola opens upwards and if the value of m is negative the graph of the parabola opens downwards.
If $\left| m \right|<1$ the graph of the parabola widens and if $\left| m \right|>1$ the graph of the parabola becomes narrower.
Example:
The vertex form of the quadratic equation ${{x}^{2}}+6x+9$ is given by ${{\left( x+3 \right)}^{2}}+0$
The factored form of a quadratic equation is given as follows,
$\Rightarrow y=\left( ax+b \right)\left( cx+d \right)$
Here,
a, c are the numerical coefficients of the term x
b, d are the constant terms
Example:
The factored form of the quadratic equation ${{x}^{2}}+6x+9$ is given by $\left( x+3 \right)\left( x+3 \right)$

Note: The given question is a direct formula based and there is no trick used to solve the question. Any mistake in writing the representation of the standard form, vertex form, factored form of a quadratic equation will result in an incorrect solution.