Question

How do you solve \[x(1 - x) + 2x - 4 = 8x - 24 - {x^2}\]?

Answer 1

Hint: Solve the given equation i.e. bring all coefficients together for the same variables. Cancel possible terms and write the equation in simplest form. Equate the equation to 0 in order to simplify the equation.

Complete step-by-step answer:
We are given the equation \[x(1 - x) + 2x - 4 = 8x - 24 - {x^2}\]
Since the equation has only one variable i.e. x, we will calculate the value of x.
Multiply the terms outside the bracket with terms inside the bracket on left hand side of the equation
\[ \Rightarrow x \times 1 - x \times x + 2x - 4 = 8x - 24 - {x^2}\]
Calculate each product on left hand side of the equation
\[ \Rightarrow x - {x^2} + 2x - 4 = 8x - 24 - {x^2}\]
Shift all values to left hand side of the equation
\[ \Rightarrow x - {x^2} + 2x - 4 - 8x + 24 + {x^2} = 0\]
Cancel possible terms from left hand side of the equation
 \[ \Rightarrow x + 2x - 4 - 8x + 24 = 0\]
Group terms with same coefficients together on left hand side of the equation
\[ \Rightarrow \left( {x + 2x - 8x} \right) + (24 - 4) = 0\]
Calculate the value inside the brackets
\[ \Rightarrow - 5x + 20 = 0\]
To simplify the equation, we will equate the equation to 0
We equate \[ - 5x + 20 = 0\]
Bring all constant values to right hand side of the equation
\[ \Rightarrow - 5x = - 20\]
We can write right hand side of the equation as product of factors of the number i.e. \[ - 20 = 4 \times ( - 5)\]
 \[ \Rightarrow - 5x = - 5 \times 4\]
Cancel same factors from both sides of the equation i.e. -5
\[ \Rightarrow x = 4\]

\[\therefore \]Solution of the equation \[x(1 - x) + 2x - 4 = 8x - 24 - {x^2}\] is \[x = 4\]

Note:
Many students make the mistake of thinking of the equation as a quadratic equation as there is the term \[{x^2}\]on both sides of the equation. Keep in mind the same values can be cancelled from both sides of the equation directly or can be brought to one side of the equation and then subtracted. Also, keep in mind when shifting values from one side of the equation to another side of the equation, always change sign from positive to negative and vice-versa.