🧮 algebra

1. The problem asks to add the fractions $\frac{1}{3}$ and $\frac{1}{2}$. To add fractions, we need a common denominator. 2. The least common denominator (LCD) of 3 and 2 is 6.

1. **Problem 1: Multiply the fractions** $\frac{2}{3} \times \frac{6}{5}$. 2. Use the multiplication rule for fractions: multiply numerators and denominators:

1. **State the problem:** We need to factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers t

1. The problem is to find the number that replaces the box in the equation $\frac{3}{4} = \frac{\Box}{20}$. 2. We use the property of equivalent fractions: if $\frac{a}{b} = \frac{

1. **State the problem:** Solve the equation $3^{1-2x} = 4^x$ for $x$. 2. **Recall the formula and rules:** When bases are different, take the logarithm of both sides to solve for

1. The problem is to verify the equation $1=1$. 2. This is a simple equality statement asserting that the number one is equal to itself.

1. The problem asks: In 2015, how many more dollars was Apple's stock price compared to its price in 2010? 2. From the graph description, the stock price in 2010 was about $30.

1. **State the problem:** We are given the function $f(x) = -5^x$ and asked to find its vertical and horizontal asymptotes. 2. **Recall the definitions:**

1. **State the problem:** We have 11 ducks, each having either 8 or 9 ducklings. The total number of ducklings is 92. We need to find how many ducks had 8 ducklings. 2. **Define va

1. **State the problem:** Simplify the expression $$\frac{\left(\frac{2}{3} a^{9} b^{-10} c^{4}\right)^{-1} \left(\frac{5}{4} a^{-2} b^{3} c^{-5}\right)^{-2}}{\left(-\frac{1}{3} a^

1. **Problem Statement:** Factor the polynomial using the Remainder Theorem. 2. **Remainder Theorem:** If a polynomial $f(x)$ is divided by $(x - a)$, the remainder is $f(a)$.

1. **State the problem:** Solve the inequality $$\frac{x^2 - 3x - 4}{x + 1} < 0$$. 2. **Factor the numerator:** The quadratic $x^2 - 3x - 4$ factors as $$(x - 4)(x + 1)$$.

1. **State the problem:** Solve the equation $\left(\log_3 x\right)^2 - 3\log_3 x = 10$ for $x$. 2. **Use substitution:** Let $y = \log_3 x$. The equation becomes:

1. **State the problem:** Solve the system represented by the augmented matrix $$\left[\begin{array}{cc|c} 1 & -2 & 7 \\ 0 & 0 & -9 \\ \end{array}\right]$$

1. **State the problem:** We are given the quadratic equation $$y = (x + 2)(x - 8)$$ and asked to sketch its graph by understanding its key features. 2. **Rewrite the equation:** E

1. **State the problem:** Find the coordinates where the graph of the quadratic equation $$y = (4 - x)(x + 5)$$ crosses the x-axis and y-axis. 2. **Find x-intercepts:** The graph c

1. The problem is to find the three answers, which typically means solving a cubic equation or finding three roots of a given problem. 2. If the problem is to solve a cubic equatio

1. **State the problem:** Find the points where the quadratic graph $y = 5x^2 + 23x + 12$ crosses the $x$-axis and $y$-axis. 2. **Recall the rules:**

1. **Stating the problem:** We need to write equations for two lines based on their graphs. 2. **Understanding the graphs:**

1. **State the problem:** Factor the quadratic equation $$-x^2 + 3x + 10 = 0$$. 2. **Rewrite the equation:** It is easier to factor if the leading coefficient is positive. Multiply

1. **State the problem:** Solve the quadratic equation $-x^2 + 3x + 10 = 0$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $-1$ to get a standard form: