🧮 algebra

1. Statement of the problem. Suppose a sample of a substance decayed to 77.8% of its original amount after 300 days, and we are asked to find the half-life and the time to reach on

1. **Problem:** If the roots of the equation $ax^2 + cx + c = 0$ are in the ratio $p : q$, show that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0.$$ **Solutio

1. The problem is to factor the expression $x^2$. 2. Notice that $x^2$ is a perfect square, which means it can be written as $x \times x$.

1. **State the problem:** We have a circle given by the equation $$x^2 + y^2 - 10x + 8y + 25 = 0$$ and a line given by $$y = kx$$. We want to find the values of $k$ for which the l

1. **State the problem:** Solve the system of linear equations: $$1.2y + 10.4 = -8.2x$$

1. **State the problem:** Solve the system of equations: $$3x - 4y = 23 \quad (1)$$

1. **Problem statement:** (a) Given a rectangular garden with area 80 m², width $y$ metres, length $x$ metres, and a path of uniform width 2 metres around it.

1. Bentuk sederhana dari $\left(6^{-2} a\right)^3 : \left(12^2 a^3\right)^{-2}$ adalah... Langkah-langkah:

1. The problem is to simplify the expression $2\ln(e^3) + \ln(e^2)$.\n\n2. Recall the logarithm power rule: $\ln(a^b) = b\ln(a)$. Since $\ln(e) = 1$, we can simplify each term:\n\n

1. Bentuk sederhana dari $\frac{6^{-2} a^3}{(12^1 a^3)^{-2}}$ adalah... Langkah-langkah:

1. The problem is to analyze the function $f(n) = \frac{n}{20 - 4n}$.\n\n2. First, identify the domain. The denominator cannot be zero, so solve $20 - 4n = 0$ which gives $n = 5$.

1. **State the problem:** Solve the equation $$\frac{4r-3}{6r+1} = \frac{2r-1}{3r+4}$$ for $r$. 2. **Cross-multiply to eliminate the fractions:**

1. The problem is to find the domain of a function, but the function is not specified. 2. The domain of a function is the set of all possible input values (x-values) for which the

1. The problem is to find the domain of the function $$f(n) = \frac{n}{20} - 4n$$. 2. The function is a rational expression where the numerator is $$n$$ and the denominator is $$20

1. We are given the equation $$\frac{x^{28}}{x^w} = x^7$$ and need to find the value of $w$. 2. Recall the law of exponents for division: $$\frac{x^a}{x^b} = x^{a-b}$$.

1. The problem is to fully simplify the expression $7 \times v^2 - 4v \times v$. 2. First, rewrite the expression using standard algebraic notation: $$7v^2 - 4v \cdot v$$.

1. The problem shows a number pyramid with algebraic expressions. The top brick is $6h^2 + 9h + 4$. 2. The second row has two bricks: left is $3h^2 + 5h$, right is unknown (empty).

1. **State the problem:** Simplify the expression $9d + 5n - 6d - 2n + 11 - 11 - 3d$. 2. **Group like terms:** Group the terms with $d$, $n$, and constants separately:

1. The problem is to rearrange the formula $h = a v$ to make $a$ the subject. 2. Start with the equation:

1. **State the problem:** We are given a parabola with vertex at $(3,1)$ and need to find the x-intercepts $p$ and $q$ of the parabola. 2. **Identify the vertex form of the parabol

1. **State the problem:** Make $h$ the subject of the equation $h + p = w$. 2. **Isolate $h$:** To make $h$ the subject, we need to get $h$ alone on one side of the equation.