Question

How do you write the quadratic equation in standard form
\[{x^2} - 12 + 4x = {\left( {x - 2} \right)^2} + 2\]

Answer 1

Hint: A quadratic equation is a second degree polynomial equation having a standard form of form $a{x^2} + bx + c$. By expanding the both sided terms of the equation we will have the required result.

Complete step by step answer:
The given equation is \[{x^2} - 12 + 4x = {\left( {x - 2} \right)^2} + 2\]
First step is to expand the bracket or solve the brackets to get rid of the squared power. We need to get rid of the squared power because all other terms have power $1$. It is always a rule that to simply the equation, it is necessary to bring the terms in the same exponents.
\[ \Rightarrow {x^2} - 12 + 4x = {x^2} - 4x + 4 + 2\]
On adding RHS, we get
\[ \Rightarrow {x^2} - 12 + 4x = {x^2} - 4x + 6\]
Bringing all variable terms on one side and the constant terms on side we get,
\[ \Rightarrow {x^2} + 4x - {x^2} + 4x = 6 + 12\]
On adding RHS, we get
\[ \Rightarrow {x^2} + 4x - {x^2} + 4x = 18\]
Adding and subtracting like terms we get
\[ \Rightarrow 8x = 18\]
Therefore, standard form of
\[{x^2} - 12 + 4x = {\left( {x - 2} \right)^2} + 2\]
Open the brackets, we get
\[8x = 18\]
By arranging in the standard form
$8x - 18 = 0$
Where coefficient of ${x^2} = 0$.

Note: Before solving a quadratic equation using the Quadratic Formula, it's vital that you be sure the equation is in this form. If you don't, you might use the wrong values for $a,b$ or $c$, and then the formula will give incorrect solutions.
Quadratic equations are the polynomial equations of degree $2$ in one variable of type \[f(x) = a{x^2} + bx + c = 0\] where \[a,b,c \in R\] and \[a \ne 0\] . It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of \[f(x)\]. The values of x satisfying the quadratic equation are the roots of the quadratic equation \[\left( {\alpha ,\beta } \right)\].
The quadratic equation always has two roots. The nature of roots may be either real or imaginary.
A quadratic polynomial, when equated to zero, becomes a quadratic equation. The values of x satisfying the equation are called the roots of the quadratic equation.
The values of variables satisfying the given quadratic equation are called its roots. In other words, \[x = \alpha \] is a root of the quadratic equation \[f(x)\], if \[f(\alpha ) = 0\].