🧮 algebra

1. **Problem Statement:** The present age of the father is the square of the present age of his son. We need to find after how many years the son's age will be half of the present

1. **State the problem:** Find the domain of the function $f(x) = x^3$. 2. **Recall the definition of domain:** The domain of a function is the set of all possible input values ($x

1. **State the problem:** We are given the quadratic function $$y = -\frac{1}{16}(x-6)^2 + 7$$ and we want to analyze its properties. 2. **Formula and explanation:** This is a quad

1. **State the problem:** Solve the inequality $2x^2 - 8 \ge 0$. 2. **Rewrite the inequality:** Factor out the common factor 2:

1. **Stating the problem:** Solve the inequality $x^2 + x - 6 \ge 0$. 2. **Formula and rules:** To solve quadratic inequalities, first find the roots of the quadratic equation $x^2

1. **State the problem:** Solve the inequality $x^2 - x - 2 \ge 0$. 2. **Formula and rules:** To solve quadratic inequalities, first find the roots of the quadratic equation $x^2 -

1. **Stating the problem:** Solve each linear equation for $x$. 2. **General approach:** Use the distributive property to expand parentheses, combine like terms, and isolate $x$ on

1. **Stating the problem:** We have a polynomial $$P(x) = x^4 - 6x^3 + ax^2 + bx + c$$ with conditions:

1. The problem involves simplifying and understanding two sets of polynomials given as: Top-left set: (d) $16x + 24 + 2x^2$, (a) $35x^2 + 43x + 12$, (b) $-21x^2 + 58x - 21$, (c) $1

1. **Stating the problem:** Vi skal løse ligninger med parenteser, hvor vi enten kan ophæve plusparenteser direkte eller minusparenteser ved at ændre fortegn. 2. **Regler:**

1. We are given two sets of quadratic expressions and asked to analyze or simplify them. 2. The general form of a quadratic expression is $ax^2 + bx + c$.

1. **State the problem:** We are given the polynomial $$2x^4 - 9x^3 + 6x^2 + 11x - 6$$ and want to express it in the form $$(x - 1)(Ax^3 + Bx^2 + Cx + D)$$ to find constants $A$, $

1. **Stating the problem:** Solve the linear equations with multiple terms for $x$. 2. **Formula and rules:** To solve equations like these, combine like terms (all $x$ terms toget

1. **State the problem:** We need to divide the cubic polynomial $$2x^3 + 2$$ by the binomial $$2x + 2$$ and express the quotient in the form $$Ax^2 + Bx + C$$. 2. **Recall the div

1. **Problem Statement:** Understand and apply the Laws of Indices including fractional indices. 2. **Laws of Indices:**

1. **State the problem:** Simplify the expression $$\frac{2x^3 - 7x^2 + 9}{2x - 3}$$ into the form $$Ax^2 + Bx + C + \frac{R}{2x - 3}$$ where $A$, $B$, $C$, and $R$ are constants.

1. **State the problem:** Simplify the expression $$\sqrt{\frac{35}{20}}$$. 2. **Recall the property of square roots:** $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ for posit

1. **Stating the problem:** We need to expand and simplify each of the expressions a) to f) by multiplying the parentheses and then compare the results to find which expressions ar

1. **State the problem:** We need to divide the cubic polynomial $$x^3 + 8x^2 + 19x + 12$$ by the linear polynomial $$x + 4$$ using algebraic long division to find constants $$A$$,

1. **Stating the problem:** We need to expand and simplify the products of binomials given in expressions a) through f) and then match them with the provided expressions A) through

1. **Stating the problem:** We are given two expressions: the first is a fraction $$\frac{x^2 + x - 6}{x - 3x^2 + 10}$$ and the second is an expression $$x + 3x - 2$$. We need to a