Here are the solutions to the problems: Quadratic Equations 1. Solve: $3x^2 - 7x + 2 = 0$ Step 1: Identify the coefficients $a=3$, $b=-7$, $c=2$. We can solve this by factoring. We need two numbers that multiply to $ac = 3 \times 2 = 6$ and add up to $b = -7$. These numbers are $-1$ and $-6$. Step 2: Rewrite the middle term using these numbers. $$3x^2 - x - 6x + 2 = 0$$ Step 3: Factor by grouping. $$x(3x - 1) - 2(3x - 1) = 0$$ Step 4: Factor out the common term $(3x - 1)$. $$(3x - 1)(x - 2) = 0$$ Step 5: Set each factor to zero and solve for $x$. $$3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3}$$ $$x - 2 = 0 \implies x = 2$$ The solutions are $\boxed{x = \frac{1}{3}, x = 2}$. 2. Solve using completing the square: $x^2 + 6x + 5 = 0$ Step 1: Move the constant term to the right side of the equation. $$x^2 + 6x = -5$$ Step 2: To complete the square, take half of the coefficient of $x$ (which is $6$), square it, and add it to both sides. Half of $6$ is $3$, and $3^2 = 9$. $$x^2 + 6x + 9 = -5 + 9$$ Step 3: Factor the left side as a perfect square and simplify the right side. $$(x + 3)^2 = 4$$ Step 4: Take the square root of both sides. Remember to include both positive and negative roots. $$x + 3 = \pm\sqrt{4}$$ $$x + 3 = \pm 2$$ Step 5: Solve for $x$ for both positive and negative cases. $$x + 3 = 2 \implies x = 2 - 3 \implies x = -1$$ $$x + 3 = -2 \implies x = -2 - 3 \implies x = -5$$ The solutions are $\boxed{x = -1, x = -5}$. Surds and Indices 1. Simplify: $\sqrt{50} + 3\sqrt{8}$ Step 1: Simplify each surd by finding the largest perfect square factor. $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$$ $$3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times \sqrt{4} \times \sqrt{2} = 3 \times 2 \times \sqrt{2} = 6\sqrt{2}$$ Step 2: Add the simplified surds. $$5\sqrt{2} + 6\sqrt{2} = (5 + 6)\sqrt{2} = 11\sqrt{2}$$ The simplified expression is $\boxed{11\sqrt{2}}$. 2. Simplify: $(2^3 \times 2^{-5})$ Step 1: Use the rule of indices $a^m \times a^n = a^{m+n}$. $$2^3 \times 2^{-5} = 2^{3 + (-5)}$$ Step 2: Perform the addition in the exponent. $$2^{3 - 5} = 2^{-2}$$ Step 3: Use the rule $a^{-n} = \frac{1}{a^n}$ to express the result with a positive exponent. $$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$$ The simplified expression is $\boxed{\frac{1}{4}}$. Logarithms 1. Simplify: $\log_{10}(100) + \log_{10}(0.001)$ Step 1: Evaluate each logarithm separately. For $\log_{10}(100)$: We ask, "To what power must $10$ be raised to get $100$?" $$10^2 = 100 \implies \log_{10}(100) = 2$$ For $\log_{10}(0.001)$: We ask, "To what power must $10$ be raised to get $0.001$?" $$0.001 = \frac{1}{1000} = \frac{1}{10^3} = 10^{-3}$$ $$\log_{10}(0.001) = -3$$ Step 2: Add the results. $$2 + (-3) = 2 - 3 = -1$$ The simplified expression is $\boxed{-1}$. 2. Solve: $\log_2(x) = 5$ Step 1: Convert the logarithmic equation into its equivalent exponential form. The definition of a logarithm states that if $\log_b(x) = y$, then $b^y = x$. Here, $b=2$, $y=5$, and $x$ is the unknown. $$x = 2^5$$ Step 2: Calculate the value of $2^5$. $$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$$ The solution is $\boxed{x = 32}$.