Question

Solve each equation.

$\mathbf{5}\left(t^2-3t+2\right)\mathbf{=}\mathit{t}\left(5t-2\right)$

Short answer

The solution of the equation is$t=\frac{10}{13}$

Step-by-step solution

Step 1. State the concept of multiplying a polynomial by a monomial.

To find the product of a polynomial and a monomial use the distributive property. The distributive property is given by:

$\begin{array}{l}a\left(b+c\right)=a\times b+a\times c\\ \left(a+b\right)c=a\times c+b\times c\end{array}$

Step 2. Apply the distributive property. 

First, simplify both sides of the equation by applying the distributive property.

$\begin{array}{l}5\left(t^2-3t+2\right)=t\left(5t-2\right)\\ 5t^2-5\left(3t\right)+5\left(2\right)=t\left(5t\right)-t\left(2\right)\end{array}$

Step 3. Simplify for t.

Further, simplify the expression for t.

$\begin{array}{c}5t^2-15t+10=5t^2-2t\\ 5t^2-5t^2+10=-2t+15t\\ 0+10=13t\\ t=\frac{10}{13}\end{array}$

Therefore,$t=\frac{10}{13}$