🧮 algebra

1. **Problem 34:** Find $q$ for the parabola $y = x^2 + px + q$ given that it touches the $x$-axis at $x=5$. 2. Since the parabola touches the $x$-axis at $x=5$, it means $x=5$ is

1. **State the problem:** We want to understand how to graph the function $f(x) = x^2 - 2x + 1$. 2. **Formula and rules:** This is a quadratic function of the form $f(x) = ax^2 + b

1. **Masala bayoni:** 21-masala: $y = -6x^2 + 7x - 2$ kvadrat funksiyaning nollari yig'indisini toping. 2. **Formulalar va qoidalar:** Kvadrat tenglama $ax^2 + bx + c = 0$ ning ild

1. **State the problem:** We need to find the value of $x$ such that $f(x) = 1 - e^x = 0$. 2. **Set the equation to zero:**

1. **Problem:** Solve and analyze the quadratic equation $x^2 - 25 = 0$. 2. **Formula and rules:** The quadratic equation is $ax^2 + bx + c = 0$. The axis of symmetry is $x = -\fra

1. **Problem Statement:** Find the axis of symmetry, vertex, y-intercept, and x-intercepts of the quadratic function $y = ax^2 + bx + c$. Also, show how to find the x-intercepts an

1. Solve $x^2 - 25 = 0$. - This is a difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

1. **State the problem:** Calculate the volume $V$ given by the formula $V = (3.1416)(5)(9)$. 2. **Formula used:** This is a simple multiplication problem where volume $V$ is the p

1. The problem is to understand what arithmetic and geometric sequences are. 2. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is co

1. Let's start by understanding what a sequence is. A sequence is an ordered list of numbers following a specific pattern. 2. For example, the sequence $2, 4, 6, 8, \dots$ increase

1. **State the problem:** Solve for $x$ in the equation $$8.6 + 5.6x - 1.5 = 40.7.$$\n\n2. **Simplify the equation:** Combine like terms on the left side: $$8.6 - 1.5 = 7.1,$$ so t

1. **بيان المسألة:** حل المثال الأول (3.5):

1. **Stating the problem:** Find the value of $x$ that satisfies the equation $$3^{2 - x} = 27.$$

1. The problem is to increase 2000 by 50%. 2. To increase a number by a percentage, use the formula: $$\text{New value} = \text{Original value} + \left(\frac{\text{Percentage}}{100

1. The problem is to simplify or understand the expression $a^2 + b^2 - ab$. 2. This expression is a quadratic form in terms of $a$ and $b$.

1. The problem asks to find the value of $b^3$ when $b = -3$. 2. The formula for raising a number to a power is $b^3 = b \times b \times b$.

1. **State the problem:** Find the highest common factor (HCF) of $y^6$ and $y^9$. 2. **Recall the formula:** The HCF of two powers with the same base is the base raised to the min

1. **State the problem:** Solve the equation $4 - 2x + 8 = -6$ for $x$. 2. **Combine like terms on the left side:** $4 + 8 = 12$, so the equation becomes:

1. **Stating the problem:** Given the operation $a * b = a + b - 1$, we want to understand how this operation works and possibly find some properties or results using it. 2. **Unde

1. **State the problem:** Given $m = -3$ and $n = 2$, find the product $mn$. 2. **Formula used:** The product of two numbers $m$ and $n$ is given by:

1. **State the problem:** Find the intersection point of the two lines given by the equations derived from the inequalities: $$3x - 5y = 2$$