Question

How do you solve it $2{(x - 3)^2} = 8$ ?

Answer 1

Hint: Expand the equation. Write $(x - 3)$ twice and then multiply and then multiply the whole equation which is on the L.H.S with $2$. After evaluating we will get a quadratic equation, a polynomial of degree $2$. Solve it to get $2$ values of $x$. Those $2$ values will be the solution of the given polynomial expression.

Formula used:
The roots of a quadratic equation ( $a{x^2} + bx + c = 0$ ) can be found by the formula, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step-by-step answer:
Given expression, $2{(x - 3)^2} = 8$
Firstly, expand the square term.
$ \Rightarrow 2(x - 3)(x - 3) = 8$
Multiply both the polynomials.
$ \Rightarrow 2\left[ {x(x - 3) - 3(x - 3)} \right] = 8$
Evaluate further,
$ \Rightarrow 2({x^2} - 3x - 3x + 9) = 8$
$ \Rightarrow 2({x^2} - 6x + 9) = 8$
Now multiply the constant $2$ with all the terms.
$ \Rightarrow 2{x^2} - 12x + 18 = 8$
Bring the R.H.S term to L.H.S
$ \Rightarrow 2{x^2} - 12x + 18 - 8 = 0$
$ \Rightarrow 2{x^2} - 12x + 10 = 0$
Now divide the entire equation with $2$
$ \Rightarrow $${x^2} - 6x + 5 = 0$
Now, use the roots for a quadratic expression formula which is,
Considering the given Quadratic equation is in the form, $a{x^2} + bx + c = 0$
Then, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here, $a = 1;b = - 6;c = 5$
On substituting in the formula,
$ \Rightarrow x = \left[ {\dfrac{{ - ( - 6) \mp \sqrt {{{( - 6)}^2} - 4 \times 1 \times 5} }}{{2 \times 1}}} \right]$
Solve the expressions present in the root first because it has higher precedence.
$ \Rightarrow x = \left[ {\dfrac{{6 \pm \sqrt {36 - 20} }}{2}} \right]$
Evaluate further.
$ \Rightarrow x = \left[ {\dfrac{{(6) \pm \sqrt {16} }}{2}} \right]$
$ \Rightarrow x = \left[ {\dfrac{{6 \pm 4}}{2}} \right]$
Write $2$ terms, one for $ + $ and another for $ - $;
$ \Rightarrow x = \left[ {\dfrac{{6 + 4}}{2}} \right];x = \left[ {\dfrac{{6 - 4}}{2}} \right]$
$ \Rightarrow x = \left[ {\dfrac{{10}}{2}} \right];x = \left[ {\dfrac{2}{2}} \right]$
$\therefore x = 5;x = 1$
Likewise, we got two roots for an equation of degree of $2$

$\therefore $The solution for the expression $2{(x - 3)^2} = 8$ is $x = 5;x = 1$

Additional information: A polynomial is a mathematical expression that contains one or more variables in sum or subtraction format with different powers. Whenever there is a polynomial that is to be solved, the solution contains the roots of the expression. The number of roots is decided by the degree of the polynomial.

Note:
Never forget to take two conditions whenever there is a $ \pm $ symbol. The expression is written twice once with $ + $ and another time with $ - $. An equation is known as a linear equation if the degree of the expression or polynomial is $1$. The solution for a linear equation is also $1$ a term. Always use the BODMAS rule while expanding any expression.