🧮 algebra

1. The problem is to expand the expression $ (x-5)^2 $. 2. Recall the formula for the square of a binomial:

1. **State the problem:** Expand the expression $$(2x+3)(5x-2)$$. 2. **Apply the distributive property (FOIL method):** Multiply each term in the first parenthesis by each term in

1. **State the problem:** We are given a line $l$ with slope $-1$. We need to compare Quantity A, the x-intercept of line $l$, and Quantity B, the y-intercept of line $l$. 2. **Rec

1. The problem is to simplify the expression $2^{3x} \div 2^x$. 2. Recall the property of exponents: $\frac{a^m}{a^n} = a^{m-n}$ when the base $a$ is the same.

1. The problem is to simplify the expression $\frac{3^{2x}}{3^x}$.\n\n2. Recall the property of exponents: $\frac{a^m}{a^n} = a^{m-n}$.\n\n3. Apply this property to the given expre

1. Problem 36: Find the smallest natural number in 25 consecutive natural numbers whose sum is 1000. Let the smallest number be $x$. Then the numbers are $x, x+1, x+2, \ldots, x+24

1. **State the problem:** Solve the equation $$\log_x 2 \cdot \log_{\frac{x}{16}} 2 = \log_{\frac{x}{64}} 2.$$\n\n2. **Rewrite the logarithms in terms of a common base:** Let $$a =

1. The problem is to find the time $t$ in the simple interest formula $$A = P(1 + rt)$$ given $A = 2867.30$, $P = 1234.10$, and $r = 0.66\% = 0.0066$. 2. Substitute the known value

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 2I_1 - I_2 + 3I_3 + 4I_4 = 9 \\ I_1 - 2I_3 + 7I_4 = 11 \\ 3I_1 - 3I_2 + I_3 + 5I_4 = 8 \\ 2I_1 + I_2

1. **Stating the problem:** A manufacturer must increase the height of a rectangular wooden door by $\frac{1}{6}$ of its original height while keeping the thickness the same. To ma

1. The problem is to find the length of the diameter of the circle given by the equation $x^2 + y^2 - 6x + 14y = 6$. 2. First, rewrite the equation in standard form by completing t

1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 + 18x + 14y + 105 = 0$$

1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 + 10x + 12y + 25 = 0$$

1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 + 6y - 72 = 0$$

1. **State the problem:** Solve the system of equations using Gauss-Jordan elimination: $$\begin{cases} -y + 2z - w = -1 \\ 2x + y - 2z - 2w = -2 \\ -x + 2y - 4z + w = 1 \\ 3x - 3w

1. **State the problem:** Simplify the expression $$\sqrt{(X+Y)(X-Y)(X^2+Y^2)(X^4+Y^4)} + Y^8$$. 2. **Simplify inside the square root:** Notice that $$(X+Y)(X-Y) = X^2 - Y^2$$ by t

1. The problem involves analyzing the points where the vertical brown line intersects the blue curve on the graph. 2. The given x-values are $x=\frac{1}{2}$, $x=0$, $x=1$, and $x=-

1. The problem presents a comparison of prices for five items at two different stores: Super Groceries and Convenient Grocers. 2. We want to analyze the price differences for each

1. The problem is to determine if the proportion $$\frac{4}{28} = \frac{5}{35}$$ is true or false. 2. Simplify each fraction separately.

1. The problem is to simplify the fraction $\frac{91}{5}$.\n\n2. Since 91 and 5 have no common factors other than 1, the fraction is already in its simplest form.\n\n3. We can also

1. Stating the problem: Solve the equation $$\frac{5}{7}x - 4 = 9$$ for $x$. 2. Add 4 to both sides to isolate the term with $x$: