Question

How do you evaluate $\left( 2y-3 \right)\left( 5y-3 \right)$?

Answer 1

Hint: Assume the given expression as ‘E’. Multiply each term of the binomial expression $\left( 2y-3 \right)$ with each term of the other binomial expression $\left( 5y-3 \right)$ to simplify the expression. Use the formula of exponents given as ${{a}^{m}}\times {{a}^{m}}={{a}^{m+n}}$ to evaluate the exponent of y and form a quadratic expression in y.

Complete step-by-step answer:
Here, we have been provided with the expression $\left( 2y-3 \right)\left( 5y-3 \right)$ and we are asked to evaluate it. That means we have to multiply the two binomial terms. Let us assume this product as ‘E’, so we have,
$\Rightarrow E=\left( 2y-3 \right)\left( 5y-3 \right)$
Now, we can clearly see that we have two linear expressions in y, $\left( 2y-3 \right)$ and $\left( 5y-3 \right)$, and both of them contain two terms, so we can say that they are binomial terms. Therefore, to simplify their product we need to multiply each term of the first binomial $\left( 2y-3 \right)$ with each term of the second binomial $\left( 5y-3 \right)$. So considering their product we get,
\[\begin{align}
  & \Rightarrow E=\left( 2y-3 \right)\times 5y-\left( 2y-3 \right)\times 3 \\
 & \Rightarrow E=\left( 2y\times 5y \right)-\left( 3\times 5y \right)-\left[ \left( 2y\times 3 \right)-3\times 3 \right] \\
\end{align}\]
Using the formula of exponent ${{a}^{m}}\times {{a}^{m}}={{a}^{m+n}}$, we get,
\[\begin{align}
  & \Rightarrow E=10{{y}^{2}}-15y-\left[ 6y-9 \right] \\
 & \Rightarrow E=10{{y}^{2}}-15y-6y+9 \\
 & \Rightarrow E=10{{y}^{2}}-21y+9 \\
\end{align}\]
Hence, the above quadratic expression is our answer.

Note: One may note that here we are getting a quadratic expression as our answer because there were two linear expressions in y and we were asked to take their product. In these types of questions you must be careful about the sign between the two terms. Remember the algebraic relations: $\left( -1 \right)\times 1=-1$ and $\left( -1 \right)\times \left( -1 \right)=1$ to simplify the expression. You must remember some important formulas of the topic exponents and powers.