Question

How do you solve \[2k(-3k+4)+6({{k}^{2}}+10)=k(4k+8)-2k(2k+5)\]?

Answer 1

Hint: We are given an expression which we have to solve for \[k\]. We will have to separate the constant terms and the \[k\] terms. We will begin by opening the brackets and multiplying the corresponding terms. We will solve further and cancel the similar terms and write the expression in terms of \[k\] and get its value.

Complete step by step answer:
According to the question, we have to solve the expression for \[k\].
So, the given expression we have is,
\[2k(-3k+4)+6({{k}^{2}}+10)=k(4k+8)-2k(2k+5)\]
We will have to separate the terms without \[k\], that is the constant terms and the terms with \[k\]. Thereby, writing the expression in terms of \[k\].
We will open the brackets and multiply the corresponding terms, we have,
\[\Rightarrow 2k(-3k)+2k(4)+6{{k}^{2}}+6(10)=k(4k)+8k-2k(2k)-2k(5)\]
Multiplying the terms, we get,
\[\Rightarrow -6{{k}^{2}}+8k+6{{k}^{2}}+60=4{{k}^{2}}+8k-4{{k}^{2}}-10k\]
Now, we will cancel out the similar terms, we get,
\[\Rightarrow 8k+60=8k-10k\]
Subtracting \[8k\] on both sides of the equality, we have,
\[\Rightarrow 8k+60-8k=8k-10k-8k\]
\[\Rightarrow 60=-10k\]
\[\Rightarrow 10k=-60\]
Now, dividing by 10 on both sides of the equality, we get the value of \[k\] as ,
\[\Rightarrow k=-6\]

Therefore, the value of \[k=-6\].

Note: We now have the value of \[k=-6\], but we don’t know if it is the correct answer or not, so we will substitute the value of \[k=-6\] back into the given expression and check if LHS = RHS.
The expression we have is,
\[2k(-3k+4)+6({{k}^{2}}+10)=k(4k+8)-2k(2k+5)\]
Taking LHS first, we have,
\[2k(-3k+4)+6({{k}^{2}}+10)\]
Substituting the value of \[k=-6\], we get,
\[\Rightarrow 2(-6)(-3(-6)+4)+6({{(-6)}^{2}}+10)\]
Multiplying the corresponding terms, we get,
\[\Rightarrow -12(18+4)+6(36+10)\]
\[\Rightarrow -12(22)+6(46)\]
\[\Rightarrow -264+276\]
On solving we get,
\[\Rightarrow 12\]
Now, we will take the RHS, we have,
\[k(4k+8)-2k(2k+5)\]
Substituting the value of \[k=-6\], we have,
\[\Rightarrow -6(4(-6)+8)-2(-6)(2(-6)+5)\]
Multiplying the terms in the above expression, we have,
\[\Rightarrow -6(-24+8)+12(-12+5)\]
\[\Rightarrow -6(-16)-12(7)\]
\[\Rightarrow 96-84\]
We get the value as,
\[\Rightarrow 12\]
So, we can see that the LHS = RHS,
Hence, we have the correct value of \[k\] as \[-6\].