Given the expression (1)/(124) x^2 - A + (1)/(4) is a perfect square.

Morning phares.m — let's get this done. Here are the solutions to your problems. Problem 1: Given the expression $\frac{1}{124} x^2 - A + \frac{1}{4}$ is a perfect square. A perfect square trinomial has the form $(px+q)^2 = p^2x^2 + 2pqx + q^2$. Comparing the given expression with this form: The coefficient of $x^2$ is $p^2 = \frac{1}{124}$. The coefficient of $x$ is $2pq$. In the given expression, there is no $x$ term, so its coefficient is $0$. Thus, $2pq = 0$. The constant term is $q^2 = -A + \frac{1}{4}$. Step 1: Use the coefficient of the $x$ term. Since $p^2 = \frac{1}{124}$, $p$ cannot be $0$. From $2pq = 0$, and knowing $p \neq 0$, it must be that $q = 0$. Step 2: Use the constant term. Substitute $q=0$ into the constant term equation: $q^2 = -A + \frac{1}{4}$ $0^2 = -A + \frac{1}{4}$ $0 = -A + \frac{1}{4}$ Step 3: Solve for $A$. $A = \frac{1}{4}$ The value of $A$ is $\boxed{\frac{1}{4}}$. --- Problem 2: Given the equation $\log_3 (x^2-9) = 3\log_2 2 + 1$. Step 1: Simplify the right-hand side (RHS) of the equation. Recall that $\log_b b = 1$. Therefore, $\log_2 2 = 1$. RHS $= 3(1) + 1 = 3 + 1 = 4$. Step 2: Rewrite the equation with the simplified RHS. $\log_3 (x^2-9) = 4$. Step 3: Convert the logarithmic equation to an exponential equation. If $\log_b y = z$, then $y = b^z$. So, $x^2-9 = 3^4$. Step 4: Calculate $3^4$. $3^4 = 3 \times 3 \times 3 \times 3 = 81$. Step 5: Solve for $x$. $x^2 - 9 = 81$ $x^2 = 81 + 9$ $x^2 = 90$ $x = \pm\sqrt{90}$ Step 6: Simplify the square root. $\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}$. So, $x = \pm 3\sqrt{10}$. Step 7: Check the domain of the logarithm. For $\log_3 (x^2-9)$ to be defined, the argument must be positive: $x^2-9 > 0$. This means $x^2 > 9$. If $x = 3\sqrt{10}$, then $x^2 = (3\sqrt{10})^2 = 9 \times 10 = 90$. Since $90 > 9$, this solution is valid. If $x = -3\sqrt{10}$, then $x^2 = (-3\sqrt{10})^2 = 9 \times 10 = 90$. Since $90 > 9$, this solution is also valid. The values of $x$ are $\boxed{\pm 3\sqrt{10}}$. Send me the next one 📸