🧮 algebra

1. The property used in expansion is the **distributive property** of multiplication over addition. 2. This property states that for any numbers $a$, $b$, and $c$, we have $$a(b+c)

1. נניח שהאיבר הראשון בסדרה החשבונית הוא $a_1 = a$ והפרש הסדרה הוא $d$. 2. מספר האיברים בסדרה הוא $3n$.

1. The problem is to understand the transformation from the base function $y=x^2$ to the transformed function $y=(x-2)^2$. 2. The base function $y=x^2$ is a parabola with its verte

1. Stating the problem: Solve the equation $$3(x + 1)^2 + 2(x - 3)^2 = (5x - 2)(x - 1) + 30$$ for $x$. 2. Expand each term:

1. The problem is to simplify the expression $\frac{5}{5_{base3}}$. 2. First, convert the base 3 number $5_{base3}$ to base 10.

1. **State the problem:** We have a function representing the cost $C$ in k10 for internet browsing based on the number of hours $h$ used. The cost function is $C(h) = 10h$. 2. **D

1. The problem is to simplify the expression $\frac{1}{3} - 5_{base\ 2} + \frac{1}{3} + 5_{base\ 2}$.\n\n2. First, recognize that $5_{base\ 2}$ is a number in base 2. However, the

1. The problem involves simplifying the expression: $\frac{1}{3} - 5 \times 2 + \frac{1}{3} + 5 \times 2$. 2. First, calculate the multiplication parts: $5 \times 2 = 10$.

1. **State the problem:** Wendy and Anna together have 136 plums. Wendy has 14 more plums than Anna. We need to find how many plums Anna has. 2. **Define variables:** Let $A$ be th

1. The problem states that the terms in the sequence increase by the same number each time, meaning it is an arithmetic sequence. 2. We are given Term 1 = 3 and Term 5 = 19.

1. The problem asks us to find which two recurring decimals among $0.3\dot{5}$, $0.35\dot{3}$, $0.35\dot{3}$, and $0.3\dot{5}$ are equivalent. 2. Let's write each recurring decimal

1. Let's analyze the behavior of the function as $x$ approaches negative infinity. 2. Suppose the function is $y = a^x$ where $a > 0$ and $a \neq 1$.

1. **State the problem:** We need to sketch a polynomial $P(x)$ with zeros at $-5$, $1$, and $4$, and with end behavior such that as $x \to \infty$, $P(x) \to -\infty$, and as $x \

1. **State the problem:** Calculate the square root of $$18 - 12\sqrt{2}$$. 2. **Rewrite the expression:** We want to find $$\sqrt{18 - 12\sqrt{2}}$$.

1. **Problem 1: Multiple Choice with Justifications** 1) Calculate $|\sqrt{5} - 2| + |2 - \sqrt{5}|$.

1. **State the problem:** Solve the equation $$|4^{y+1} - 1| = |4^y - 3|$$ correct to 3 significant figures. 2. **Rewrite the equation:** Note that $$4^{y+1} = 4 \cdot 4^y$$, so th

1. The problem is to simplify the expression $\sqrt{-x^2}$.\n\n2. Recall that $x^2$ is always non-negative for any real number $x$, so $-x^2$ is non-positive (less than or equal to

1. Problem 12: A car uses 5 liters of gasoline to travel 40 km. How much gasoline is needed to travel 128 km at the same rate? Step 1: Find the rate of gasoline consumption per km.

1. Justify that $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8})^{12} = 4\sqrt{14} - 15$$ Step 1: Recognize the conjugate terms and rewrite the expression.

1. **Justify that** $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8}) = 4 \sqrt{14} - 15$$ - Start by noting the conjugate multiplication: $$(\sqrt{7} - \sqrt{8})(\sqrt{7} + \sqrt

1. Given the system of equations: $$x + y - z = 2$$