Chapter T, Problem 3ADT

Expand and simplify.

1. (a) 3(x + 6) + 4(2x − 5)
2. (b) (x + 3)(4x − 5)
3. (c) $\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)$
4. (d) (2x + 3)²
5. (e) (x + 2)³

To determine

To simplify: The expression $3\left(x+6\right)+4\left(2x-5\right)$.

Answer to Problem 3ADT

The expression $3\left(x+6\right)+4\left(2x-5\right)$ is simplified as $11x-2$.

Explanation of Solution

Consider the expression $3\left(x+6\right)+4\left(2x-5\right)$.

Simplify the above expression as follows,

$\begin{array}{c}3\left(x+6\right)+4\left(2x-5\right)=3x+3\times 6+4\times 2x-4\times 5\\ =3x+18+8x-20\\ =11x-2\end{array}$

Thus, the expression $3\left(x+6\right)+4\left(2x-5\right)$ is simplified as $11x-2$.

To determine

To simplify: The expression $\left(x+3\right)\left(4x-5\right)$.

Answer to Problem 3ADT

The expression $\left(x+3\right)\left(4x-5\right)$ is simplified as $4x^2+7x-15$.

Explanation of Solution

Consider the expression $\left(x+3\right)\left(4x-5\right)$.

Simplify the above expression as follows,

$\begin{array}{c}\left(x+3\right)\left(4x-5\right)=x\left(4x-5\right)+3\left(4x-5\right)\\ =4x^2-5x+12x-15\\ =4x^2+7x-15\end{array}$

Thus, the expression $\left(x+3\right)\left(4x-5\right)$ is simplified as $4x^2+7x-15$.

To determine

To simplify: The expression $\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)$.

Answer to Problem 3ADT

The expression $\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)$ is simplified as $a-b$.

Explanation of Solution

Consider the expression $\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)$.

Simplify the above expression as follows,

$\begin{array}{c}\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)=\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)-\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)\\ ={\left(\sqrt{a}\right)}^2-\sqrt{a}\sqrt{b}+\sqrt{a}\sqrt{b}-{\left(\sqrt{b}\right)}^2\\ =a-b\end{array}$

Thus, the expression $\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)$ is simplified as $a-b$.

To determine

To simplify: The expression ${\left(2x+3\right)}^2$.

Answer to Problem 3ADT

The expression ${\left(2x+3\right)}^2$ is simplified as $4x^2+12x+9$.

Explanation of Solution

Consider the expression ${\left(2x+3\right)}^2$.

Simplify the above expression as follows,

$\begin{array}{c}{\left(2x+3\right)}^2={\left(2x\right)}^2+2\times 2x\times 3+3^2\left(\because {\left(a+b\right)}^2=a^2+2ab+b^2\right)\\ =4x^2+12x+9\end{array}$

Thus, the expression ${\left(2x+3\right)}^2$ is simplified as $4x^2+12x+9$.

To determine

To simplify: The expression ${\left(x+2\right)}^2$.

Answer to Problem 3ADT

The expression ${\left(x+2\right)}^2$ is simplified as $x^2+4x+4$.

Explanation of Solution

Consider the expression ${\left(x+2\right)}^2$.

Simplify the above expression as follows,

$\begin{array}{c}{\left(x+2\right)}^2={\left(x\right)}^2+2\times x\times 2+2^2\left(\because {\left(a+b\right)}^2=a^2+2ab+b^2\right)\\ =x^2+4x+4\end{array}$

Thus, the expression ${\left(x+2\right)}^2$ is simplified as $x^2+4x+4$.