If 2x + 5 = 17, find the value of x.

Here are the solutions to your assignment questions:

1a. Solve $5x^2 - 7x - 6 = 0$.

Step 1: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=5$, $b=-7$, $c=-6$. $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(5)(-6)}}{2(5)}$

Step 2: Simplify the expression under the square root. $x = \frac{7 \pm \sqrt{49 + 120}}{10}$ $x = \frac{7 \pm \sqrt{169}}{10}$

Step 3: Calculate the square root and find the two solutions. $x = \frac{7 \pm 13}{10}$ $x_1 = \frac{7 + 13}{10} = \frac{20}{10} = 2$ $x_2 = \frac{7 - 13}{10} = \frac{-6}{10} = -\frac{3}{5}$ The solutions are $\boxed{x = 2, -\frac{3}{5}}$.

1b. Solve $4x^2 + 2x + 3 = 0$.

Step 1: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=4$, $b=2$, $c=3$. $x = \frac{-2 \pm \sqrt{2^2 - 4(4)(3)}}{2(4)}$

Step 2: Simplify the expression under the square root. $x = \frac{-2 \pm \sqrt{4 - 48}}{8}$ $x = \frac{-2 \pm \sqrt{-44}}{8}$

Step 3: Simplify the square root involving the imaginary unit $i$. $x = \frac{-2 \pm \sqrt{4 \times -11}}{8}$ $x = \frac{-2 \pm 2i\sqrt{11}}{8}$

Step 4: Simplify the fraction. $x = \frac{2(-1 \pm i\sqrt{11})}{8}$ $x = \frac{-1 \pm i\sqrt{11}}{4}$ The solutions are $\boxed{x = \frac{-1 \pm i\sqrt{11}}{4}}$.

2a. Convert $101101_2$ to base ten.

Step 1: Write out the place values for each digit in base 2. $101101_2 = 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$

Step 2: Calculate the value of each term. $= 1 \times 32 + 0 \times 16 + 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1$ $= 32 + 0 + 8 + 4 + 0 + 1$

Step 3: Sum the values. $= 45$ The base ten equivalent is $\boxed{45_{10}}$.

2b. Convert $156_{10}$ to base two.

Step 1: Divide 156 by 2 and record the remainder. Continue dividing the quotient by 2 until the quotient is 0. $156 \div 2 = 78 \text{ R } 0$ $78 \div 2 = 39 \text{ R } 0$ $39 \div 2 = 19 \text{ R } 1$ $19 \div 2 = 9 \text{ R } 1$ $9 \div 2 = 4 \text{ R } 1$ $4 \div 2 = 2 \text{ R } 0$ $2 \div 2 = 1 \text{ R } 0$ $1 \div 2 = 0 \text{ R } 1$

Step 2: Read the remainders from bottom to top. The base two equivalent is $\boxed{10011100_2}$.

3a. Evaluate $\sqrt{50} + \sqrt{18}$.

Step 1: Simplify each square root by finding perfect square factors. $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$ $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Step 2: Substitute the simplified terms back into the expression. $5\sqrt{2} + 3\sqrt{2}$

Step 3: Combine the like terms. $(5+3)\sqrt{2} = 8\sqrt{2}$ The evaluated expression is $\boxed{8\sqrt{2}}$.

3b. Simplify $\sqrt{12} - 2\sqrt{27}$.

Step 1: Simplify each square root. $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ $\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$

Step 2: Substitute the simplified terms back into the expression. $2\sqrt{3} - 2(3\sqrt{3})$

Step 3: Perform the multiplication and combine the like terms. $2\sqrt{3} - 6\sqrt{3} = (2-6)\sqrt{3} = -4\sqrt{3}$ The simplified expression is $\boxed{-4\sqrt{3}}$.

4a. Simplify $(3a)^3 \times (9a^5)^{-1}$.

Step 1: Apply the exponent rule $(xy)^n = x^n y^n$ to the first term. $(3a)^3 = 3^3 a^3 = 27a^3$

Step 2: Apply the exponent rule $x^{-n} = \frac{1}{x^n}$ to the second term. $(9a^5)^{-1} = \frac{1}{9a^5}$

Step 3: Multiply the simplified terms. $27a^3 \times \frac{1}{9a^5} = \frac{27a^3}{9a^5}$

Step 4: Simplify the fraction using exponent rule $\frac{x^m}{x^n} = x^{m-n}$. $= \frac{27}{9} \times a^{3-5} = 3a^{-2} = \frac{3}{a^2}$ The simplified expression is $\boxed{\frac{3}{a^2}}$.

4b. Simplify $2 - \frac{3}{x+2} + \frac{4}{(x+3)^2}$.

Step 1: Find a common denominator, which is $(x+2)(x+3)^2$. $2 - \frac{3}{x+2} + \frac{4}{(x+3)^2} = \frac{2(x+2)(x+3)^2}{(x+2)(x+3)^2} - \frac{3(x+3)^2}{(x+2)(x+3)^2} + \frac{4(x+2)}{(x+2)(x+3)^2}$

Step 2: Combine the numerators over the common denominator. $= \frac{2(x+2)(x+3)^2 - 3(x+3)^2 + 4(x+2)}{(x+2)(x+3)^2}$

Step 3: Expand $(x+3)^2 = x^2 + 6x + 9$ and then expand the numerator. $= \frac{2(x+2)(x^2+6x+9) - 3(x^2+6x+9) + 4x+8}{(x+2)(x+3)^2}$ $= \frac{2(x^3+6x^2+9x+2x^2+12x+18) - 3x^2-18x-27 + 4x+8}{(x+2)(x+3)^2}$ $= \frac{2(x^3+8x^2+21x+18) - 3x^2-18x-27 + 4x+8}{(x+2)(x+3)^2}$ $= \frac{2x^3+16x^2+42x+36 - 3x^2-18x-27 + 4x+8}{(x+2)(x+3)^2}$

Step 4: Combine like terms in the numerator. $= \frac{2x^3 + (16-3)x^2 + (42-18+4)x + (36-27+8)}{(x+2)(x+3)^2}$ $= \frac{2x^3 + 13x^2 + 28x + 17}{(x+2)(x+3)^2}$ The simplified expression is $\boxed{\frac{2x^3 + 13x^2 + 28x + 17}{(x+2)(x+3)^2}}$.

5a. Simplify $\frac{2x}{x+y} + \frac{3y}{x+y}$.

Step 1: Since the denominators are the same, add the numerators. $= \frac{2x+3y}{x+y}$ The simplified expression is $\boxed{\frac{2x+3y}{x+y}}$.

5b. If $2x+5=17$ find the value of $x$.

Step 1: Subtract 5 from both sides of the equation. $2x + 5 - 5 = 17 - 5$ $2x = 12$

Step 2: Divide both sides by 2. $\frac{2x}{2} = \frac{12}{2}$ $x = 6$ The value of $x$ is $\boxed{6}$.