Question

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

Answer 1

Hint: Firstly, evaluate the given expression by grouping the like terms to one side and then solving them. Now in the end we will be left out by a quadratic expression or a linear line equation. Solve the equation by grouping same degree terms to find the value of x. One can always crosscheck the answers by putting the value back into the expression and if in the end, we get LHS=RHS then the solution obtained is this correct.

Complete step by step solution:
The given expression is, $x\left( 1-x \right)+2x-4=8x-24-{{x}^{2}}$
The given polynomial is a polynomial of degree 2.
To solve this polynomial,
We need to group the like terms together to simplify the expression by performing mathematical operations on the same degree terms.
$\Rightarrow x-{{x}^{2}}+2x-4=8x-24-{{x}^{2}}$
$\Rightarrow \left( x+2x-8x \right)-4+24-{{x}^{2}}+{{x}^{2}}=0$
Now let us perform the mathematical operations on these like terms.
For that, we need to take the variable common to perform these mathematical operations solely on the constants.
$\Rightarrow x\left( 1+2-8 \right)-4+24-{{x}^{2}}+{{x}^{2}}=0$
Now evaluate.
$\Rightarrow x\left( -5 \right)-4+24-{{x}^{2}}+{{x}^{2}}=0$
Now coming to the ${{x}^{2}}$ terms,
Take the variable out again and then perform the mathematical operation on the coefficients.
$\Rightarrow -5x-4+24-{{x}^{2}}(1-1)=0$
Simplify further.
$\Rightarrow -5x-4+24-{{x}^{2}}(0)=0$
$\Rightarrow -5x-4+24=0$
Now perform the normal addition on the constants.
$\Rightarrow -5x+20=0$
Now subtract $20$ on both sides of the equation.
$\Rightarrow -5x+20-20=-20$
Now evaluate further.
$\Rightarrow -5x=-20$
Now divide the equation with $-5$
$\Rightarrow x=\dfrac{-20}{-5}=4$
Therefore, the solution for the expression $x\left( 1-x \right)+2x-4=8x-24-{{x}^{2}}$ is by solving the value for x which will be $4$.

Note: We can cross-check if our answer is correct by substituting our answer back into the expression.
The given polynomial is,
$\Rightarrow x\left( 1-x \right)+2x-4=8x-24-{{x}^{2}}$
On substituting the value of x as 4 we get,
$\Rightarrow 4\left( 1-4 \right)+\left( 2\times 4 \right)-4=\left( 8\times 4 \right)-24-{{4}^{2}}$
Now let us start evaluating the expression.
$\Rightarrow 4\left( -3 \right)+8-4=32-24-16$
$\Rightarrow -12+4=8-16$
$\Rightarrow -8=-8$
LHS= RHS
Hence the solution we have obtained is correct.