🧮 algebra

1. **Stating the problem:** We are given a quadratic function $y = f(x)$ with a graph of a downward-opening parabola. 2. **Given information:**

1. **Stating the problem:** We want to understand the sum of the first 8 natural numbers: $$1 + 2 + 3 + \dots + 7 + 8$$ and explore the approximations involving expressions like $$

1. **Stating the problem:** We are given the quadratic function $$y = a(x + h)^2 + k$$ with constants $a$, $h$, and $k$. The graph is a parabola opening downwards with vertex at $(

1. **Problem Statement:** We are given the quadratic function $$y = a(x + h)^2 + k$$ where $$a$$, $$h$$, and $$k$$ are constants. The graph is a parabola opening downwards with ver

1. The problem states that the graph of $y = (mx + n)^2 - 1$ passes through the origin $(0,0)$ and we need to determine which statements about $m$ and $n$ are true. 2. Since the gr

1. **State the problem:** Solve the equation $$9^x = 3^{2x+1}$$ for $x$. 2. **Rewrite the bases:** Note that $9$ can be written as $3^2$, so rewrite the equation as $$\left(3^2\rig

1. **Problem 8:** Given the polynomial $f(x) = 8px^3 + 4qx^2 - 6x$, where $p$ and $q$ are constants, and $f(x)$ is divisible by $2x + 3$. We need to find the remainder when $f(x)$

1. The problem asks to simplify the expression \( \frac{5:2}{3} \).\n2. First, interpret the colon ":" as division, so \(5:2 = \frac{5}{2}\).\n3. Now the expression becomes \( \fra

1. The problem is to simplify the expression $5 \div \frac{2}{3}$. 2. Division by a fraction is the same as multiplication by its reciprocal. The reciprocal of $\frac{2}{3}$ is $\f

1. **State the problem:** Solve the equation $$4^{x-2} = \frac{1}{8}$$ for $x$. 2. **Recall the bases:** Note that $4$ and $8$ can be expressed as powers of $2$:

1. **State the problem:** Solve the equation $$3^x = \frac{1}{27}$$ for $x$. 2. **Recall the properties of exponents:** We know that $27 = 3^3$, so $$\frac{1}{27} = 3^{-3}$$ becaus

1. **Problem Statement:** We know that when a polynomial $f(x)$ is divided by $x - 1$, the remainder is $R$. We want to find the remainder when $f(x + 1)$ is divided by one of the

1. **State the problem:** We have a polynomial $$f(x) = ax^4 + 8x^3 + 8b + 7$$ where $$a$$ and $$b$$ are constants.

1. **Problem:** In an examination, 42% of students failed in Mathematics and 52% failed in English. If 17% students failed in both subjects, find the percentage of those students w

1. The problem is to evaluate the expression $$-\sqrt{(3 - 9.69)^2}$$ given the numbers 0.144 and 9.69. 2. Recall the formula for the square root of a square: $$\sqrt{x^2} = |x|$$,

1. The problem is to evaluate the expression $3-(4,6)^2$. 2. First, clarify the notation: $(4,6)$ typically represents a point or vector, but here it seems to be a typo or misuse.

1. **Problem statement:** Given sets $A = \{3, 7, a^2\}$ and $B = \{2, 4, a+1, a+b\}$ with all elements integers, and $A \cap B = \{4, 7^2\}$, find: a) values of $a$ and $b$

1. **State the problem:** Solve the equation $x - 3 - 3x(x - 3) = 0$ for $x$. 2. **Rewrite the equation:** The equation is $x - 3 - 3x(x - 3) = 0$.

1. **Stating the problem:** Solve the equation $$x^3 - x^3 - x^3 = 0$$. 2. **Simplify the expression:** Combine like terms:

1. प्रश्न समजून घेऊया: 90 ते 180 अंकांची सरासरी आणि 80 ते 160 अंकांची सरासरी यामध्ये फरक किती आहे हे शोधायचे आहे. 2. सरासरी काढण्याचा सूत्र: $$\text{सरासरी} = \frac{\text{एकूण बेरी

1. **Problem Statement:** Solve the quadratic equation $x^2 - 5x + 6 = 0$ and sketch its graph. 2. **Formula Used:** The quadratic equation $ax^2 + bx + c = 0$ can be solved using