🧮 algebra

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:

1. **State the problem:** Find the y-intercept and x-intercept of the line given by the equation $$y = -2x + 4$$ and plot the points. 2. **Y-intercept:** The y-intercept occurs whe

1. The problem asks to find the y-intercept of the quadratic graph given by the equation $$y = (x + 4)(x - 1)$$. 2. The y-intercept of a graph is the point where the graph crosses

1. **State the problem:** Find the x-intercepts of the quadratic function given by $$y = x^2 - 2x - 15$$. 2. **Recall the formula:** The x-intercepts occur where $$y = 0$$, so solv

1. **State the problem:** We are given the system of equations: $$x + y = 13$$

1. نبدأ ببيان المسألة: حل المعادلة الجبرية أو تبسيط التعبير المعطى. 2. نستخدم القوانين الأساسية للجبر مثل التوزيع، الجمع، الطرح، الضرب، والقسمة.

1. **Problem statement:** We have a sequence representing the number of seats in each row of a hall. The first row has 9 seats, the second has 14 seats, the third has 19 seats, and

1. The problem is to simplify the expression $$\frac{12}{\sqrt{7} - \sqrt{3}} + \frac{12}{\sqrt{7} + \sqrt{3}}$$. 2. To simplify expressions with square roots in the denominator, w

1. The problem asks for the graph of the reflection of the parabola given by the equation $$y = x^2 + 3x - 10$$ over the y-axis. 2. Reflecting a graph over the y-axis means replaci

1. The problem is to explain and demonstrate the FOIL method for multiplying two binomials. 2. FOIL stands for First, Outer, Inner, Last, which are the pairs of terms you multiply