Question

How do you solve \[3t(t + 5) - {t^2} = 2{t^2} + 4t - 1\]

Answer 1

Hint: Here we have to expand the given terms by opening the brackets and try to group similar terms together. On doing some simplification we get the required answer.

Complete step by step solution:
The given equation is\[3t(t + 5) - {t^2} = 2{t^2} + 4t - 1\]
We observe that all the terms are not in their correct form.
So, the first step is always to open the brackets and get a simplified equation.
When we will open the brackets, the operation occurring would be a multiplication, i.e., will be multiplied with.
Here, we use the property of exponents according to which, if the bases are the same and they are under multiplication, the powers get added.
\[ \Rightarrow (3t \times t) + (3t \times 5) - {t^2} = 2{t^2} + 4t - 1\]
Here, we use the property of exponents according to which, if the bases are the same and they are under multiplication, the powers get added.
\[ \Rightarrow 3{t^2} + 15t - {t^2} = 2{t^2} + 4t - 1\]
Now, it is visible that all the terms are in their desired form. We need to group similar terms together in order to further simplify the equation.
Bring all the variable terms on one side and all the constant terms on one side. This is the standard way of grouping terms.
\[ \Rightarrow 3{t^2} + 15t - {t^2} - 2{t^2} - 4t = - 1\]
Grouping the similar terms we get
\[ \Rightarrow 3{t^2} - {t^2} - 2{t^2} + 15t - 4t = - 1\]
Adding and subtracting similar terms we get
\[ \Rightarrow 3{t^2} - 3{t^2} + 15t - 4t = - 1\]
Since, the power of the variable terms are same, we can operate on them
\[ \Rightarrow 0 + 11t = - 1\]
\[ \Rightarrow 11t = - 1\]
Dividing by on both the sides of the equation by \[11\] we get,
\[ \Rightarrow t = - \dfrac{1}{{11}}\]

Therefore, the simplified form of the given equation is \[t = - \dfrac{1}{{11}}\]

Note: One should always remember that if a positive term is shifted from one side of the equation to the other side, it becomes negative and vice versa. Also, the equation does not affect or change if the same quantity is added, subtracted, multiplied or divided with the terms on both sides of the equation. Hence, in this case, we divided both sides by $11$