🧮 algebra

1. The problem asks for the number of unordered pairs $(x,y)$ such that $x^2 - y^2 = 81$. 2. First, rewrite the equation using the difference of squares factorization:

1. **State the problem:** We have a 1-pound mixture of two types of rice costing 10 and 30 per pound respectively. The average cost of the mixture is 24 per pound. We need to find

1. **State the problem:** We have a polynomial $$f(x) = 2x^3 + ax^2 + bx - 6$$ and we know the remainders when divided by $$x-1$$ and $$x+4$$ are 12 and -18 respectively. We need t

1. The problem states that $x-1$ is a factor of the polynomial $f(x) = x^3 + kx^2 + 3x + 10$. We need to find the value of $k$ and then factorize the polynomial completely. 2. Sinc

1. **State the problem:** We are given that the polynomial $f[x] = 6 - x - x^2$ is a factor of $g[x] = ax^3 + 5x^2 + bx - 18$. We need to find the constants $a$ and $b$, and then f

1. **State the problem:** We have a polynomial $$f(x) = 4x^3 - 4x^2 - x - k$$ where $$k$$ is a constant. Given that $$x-1$$ is a factor of $$f(x)$$, we need to find the value of $$

1. **State the problem:** We are given that $x^2 - x - 6$ is a factor of the polynomial $f(x) = 3x^3 + px^2 + qx - 6$. We need to find the constants $p$ and $q$ and then find the z

1. **State the problem:** Solve the equation $e^{2x} = \frac{1}{2}$ for $x$. 2. **Rewrite the equation:** We have $e^{2x} = \frac{1}{2}$.

1. The problem involves understanding the polyline graph connecting the points with values 20, 10, 40, 46, and 38. 2. The polyline starts at 20, moves up to 10, then horizontally t

1. **State the problem:** Simplify the expression $$\frac{1}{7}+\frac{2}{3}+(\binom{1}{2}-2)x^2$$. 2. **Evaluate the binomial coefficient:** Recall that $$\binom{n}{k} = 0$$ if $$k

1. **Problem 8:** Describe the transformations for each graph from $y=f(x)$. - a) $y=f(x+4)$ shifts the graph of $f(x)$ 4 units to the left.

1. **State the problem:** Given the equations $x^2 = b - ax$ and $x^3(x^3 + c) = d$, find which values of $c$ and $d$ satisfy these conditions. 2. **Rewrite the first equation:** F

1. Problem 8: Describe the transformations for each graph based on $y=f(x)$. a) $f(x+4)$ shifts the graph of $f(x)$ 4 units to the left.

1. **Problem 8:** Describe transformations of the graph $y = f(x)$ for each case. a) $f(x + 4)$ shifts the graph of $f(x)$ **4 units to the left**.

1) Développer et réduire : - Pour $A = -2(2x - 3)$ :

1. The problem is to evaluate the expression $25 - 25 \div 5 \times 5$. 2. According to the order of operations (PEMDAS/BODMAS), division and multiplication are performed before su

1. The problem involves simplifying the expression $\frac{1}{6} \times \frac{1}{5}$ and understanding the value 37.5 kg. 2. First, multiply the fractions: $$\frac{1}{6} \times \fra

1. The problem asks for the coordinates of point R on the Cartesian plane. 2. The x-coordinate is the horizontal position, and the y-coordinate is the vertical position.

1. The problem is to expand the expression $5(x + \Delta x)^2$. 2. Recall the formula for the square of a binomial: $ (a + b)^2 = a^2 + 2ab + b^2 $.

1. Factor $9x^2 - 144z^2$ completely. Step 1: Recognize this as a difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

1. The problem is to analyze the linear function $y = -3x + 9$. 2. This is a straight line with slope $-3$ and y-intercept $9$.