🧮 algebra

1. **State the problem:** We are given the rational function $$f(x) = \frac{x^2 - 8x + 16}{x^2 - 16}$$ and asked to analyze and solve it. 2. **Factor numerator and denominator:**

1. **State the problem:** We are given the function $$f(x) = \frac{x^2 - 8x + 16}{x^2 - 16}$$ and need to analyze its properties including simplification, domain, asymptotes, and d

1. **State the problem:** We are given the rational function $$f(x) = \frac{x^2 - 8x + 18}{x^2 - 16}$$ and need to analyze its properties including domain, roots, and asymptotes. 2

1. Problem 59: Given the sum of the first 10 terms of an arithmetic progression (AP) is 140, find $a_2 + a_9$. - Let the first term be $a_1$ and common difference $d$.

1. Problem 1: Ike and Seb's soft drink sharing. Let Ike originally have $x$ ml and Seb have $120 - x$ ml.

1. The problem states that for any two numbers $a$ and $b$, the operation $\Delta$ is defined as $a \Delta b = (a \times b) + a$. 2. We need to find the value of $3 \Delta (4 \Delt

1. **State the problem:** We have a triangular arrangement of numbers and need to find the number that replaces the question mark (?) in the fourth row. 2. **Analyze the pattern:**

1. Simplify the dot product $(-6,3) \cdot (4,-2)$. The dot product formula is $a \cdot b = a_1b_1 + a_2b_2$.

1. The problem is to expand the expression $$(x-1)(x-9)(x-27)(x-42)$$ into a polynomial. 2. First, group the terms to make multiplication easier: $$(x-1)(x-9)$$ and $$(x-27)(x-42)$

1. Masala: 2 va 65 sonlari orasiga 20 ta son qo'yilgan, hosil bo'lgan ketma-ketlik arifmetik progressiya. Arifmetik progressiyaning o'rta arifmetigini toping. Arifmetik progressiya

1. **State the problem:** We need to find the coordinates of the points where the parabola $y = x^2 + 2$ intersects the line $y = 2x - 1$. 2. **Set the equations equal:** At the po

1. The problem involves solving the inequalities given: $$z(x+1,y) = x + y$$

1. **Evaluate** $\sqrt{\frac{11}{12}} - \frac{1}{3} + \frac{1}{2}$ without a calculator. Step 1: Simplify $\sqrt{\frac{11}{12}} = \frac{\sqrt{11}}{\sqrt{12}} = \frac{\sqrt{11}}{2\s

1. **State the problem:** Rationalise the denominator of the fraction $$\frac{8 + \sqrt{5}}{\sqrt{5} - 1}$$ and simplify the result. 2. **Multiply numerator and denominator by the

1. **State the problem:** Solve the linear equation $2x + 5x - 3 = 0$ for $x$. 2. **Combine like terms:** Add the terms with $x$ together:

1. You asked if I can draw graphs. 2. I can help explain how to graph functions and provide the equations for graphing.

1. Problem 16: What is $i^2$? Recall that $i$ is the imaginary unit defined by $i = \sqrt{-1}$.

1. Stating the problem: Solve the equation $$4r^2 - 10 = -26$$ for $$r$$. 2. Add 10 to both sides to isolate the quadratic term:

1. **Solve the equation:** $x^2 - 10 = 19$ Step 1: Add 10 to both sides to isolate $x^2$:

1. Problem 1: Given that $25x^2 - 20x + k$ is a perfect square. A quadratic is a perfect square exactly when its discriminant is zero.

1. Solve $10. |6x| + 5 = -45$. Step 1: Subtract 5 from both sides: