🧮 algebra

1. **State the problem:** Expand the expression $$\left(\frac{5}{3}x+4\right)^2$$ as a trinomial in simplest form. 2. **Formula used:** The square of a binomial $$(a+b)^2 = a^2 + 2

1. **State the problem:** We are given a graph of a function $y = f(x)$ that approaches but never touches the x-axis, passes through the y-axis at 3, and rises steeply to the point

1. **State the problem:** We need to find the equation of a line in slope-intercept form, which is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 2. **Given:** S

1. **State the problem:** We need to find which lines among the four given are parallel. 2. **Recall the rule:** Lines are parallel if and only if their slopes are equal.

1. **State the problem:** Jerome starts with a balance of $31.33 and spends $4.19 each day. We want to find how many whole days he can spend $4.19 before his balance becomes negati

1. **State the problem:** Simplify the expression $$(x2y3z)^0$$. 2. **Recall the zero exponent rule:** Any nonzero expression raised to the power of zero equals 1. That is, for any

1. **State the problem:** Simplify the expression $$(x2y3z)^0$$. 2. **Recall the zero exponent rule:** Any nonzero base raised to the power of zero equals 1. That is, for any expre

1. **State the problem:** We need to find the value of $k$ in the equation $k - 3 - 9 = 12$. 2. **Combine like terms:** Simplify the left side by combining $-3$ and $-9$:

1. **State the problem:** Factor the quadratic expression $x^2 - 7x + 10$. 2. **Recall the factoring formula:** For a quadratic $x^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** We want to find which expression is equivalent to $24x$ and then solve $24x + 8$. 2. **Check each option for equivalence to $24x$:**

1. The problem is to understand or simplify the expression $24x+8$. 2. This is a linear algebraic expression consisting of two terms: $24x$ and $8$.

1. **State the problem:** We need to find which expression is equivalent to $24x$ from the given options. 2. **Recall the distributive property:** $a(b + c) = ab + ac$. This helps

1. **State the problem:** Factor the expression $$x^3 - 5x$$. 2. **Recall the factoring rule:** For expressions with a common factor, factor it out. Here, both terms have a common

1. **State the problem:** Solve the equation $$2x = 2 \times |x| - 2$$ for $$x$$. 2. **Understand the absolute value:** Recall that $$|x| = \begin{cases} x & \text{if } x \geq 0 \\

1. Let's analyze the expressions you wrote and see if they are correct. 2. You wrote: $$2^{6} + 12 = 0$$. This is not true because $$2^{6} = 64$$, and $$64 + 12 = 76 \neq 0$$.

1. Let's clarify the question: You want to know if you can square an exponential expression to isolate $x$ and then continue solving the problem. 2. Consider an exponential equatio

1. **Problem 1:** Find the sum of the real solutions to the equation $$2^x - (\sqrt{2})^{x+6} + 12 = 0.$$ 2. **Rewrite the equation:** Note that $\sqrt{2} = 2^{\frac{1}{2}}$, so

1. **State the problem:** We are given two equations involving powers of 8 and 27 with the same exponent $x$: $$8^x + 27^x = \frac{13}{36}$$

1. **Problem 1: Find the mean of $\log_{10} 27$, $\log_{10} 64$, and $\log_{10} 216$.** 2. The mean of logarithms is the average: $$\text{mean} = \frac{\log_{10} 27 + \log_{10} 64

1. **State the problem:** We need to find the value of the quadratic polynomial $$7s^2 - 4s + 6$$ for three different values of $s$: $0$, $-3$, and $4$.

1. **State the problem:** We want to find the number of integer values of $k$ in the range $-6 \leq k \leq 6$ such that the equation $$4^x - k2^x + 4 = 0$$ has two distinct real so