🧮 algebra

1. **State the problem:** We have two polynomials:

1. **Stating the problem:** We have two polynomials:

1. Stating the problem: Simplify the expression $$(4b - \frac{1}{2}c)(5b + \frac{1}{2}c)$$. 2. Use the distributive property (FOIL method) to expand the product:

1. The problem is to subtract the fractions $\frac{3}{4}$ and $\frac{1}{4}$.\n\n2. Since both fractions have the same denominator, we can subtract the numerators directly: $$\frac{

1. **State the problem:** We need to find the base and height of a triangle where the base length is four more than twice the height, and the area is 24 m². 2. **Define variables:*

1. The problem asks to find the value of the expression $12x - 7$ when $x = 4$. 2. Substitute $x = 4$ into the expression:

1. The problem is to simplify or understand the expression $12x - 7$. 2. This is a linear expression in terms of $x$, where $12$ is the coefficient of $x$ and $-7$ is a constant te

1. Expand $\log(6 \cdot 11)$ using the product property: $\log(6) + \log(11)$. 2. Expand $\log(5 \cdot 3)$: $\log(5) + \log(3)$.

1. The problem is to simplify the expression $1 \times (m+1) \times s$. 2. Multiplying by 1 does not change the value, so the expression simplifies to $(m+1)s$.

1. The problem involves understanding the expression with a denominator containing $s^{k(m+1)}$ or $s^{k^{m+1}}$. 2. Let's clarify the expression: if the denominator is $s^{k(m+1)}

1. **State the problem:** Find the equation of the line passing through the points $(-8,4)$ and $(-8,2)$.\n\n2. **Calculate the slope:** The slope $m$ is given by $$m=\frac{y_2 - y

1. **State the problem:** Find the equation of the line passing through the point $(5,3)$ with slope $m = -\frac{1}{5}$.\n\n2. **Recall the point-slope form of a line:** The equati

1. **State the problem:** Find the equation of the line passing through the point $(-8,4)$ with slope $m = -\frac{1}{2}$.\n\n2. **Recall the point-slope form of a line:** The equat

1. **State the problem:** Show that $x+2$ is a factor of $h(x) = x^3 - x^2 - 24x - 36$ and then factor $h(x)$ completely. 2. **Check if $x+2$ is a factor using the Factor Theorem:*

1. The quadratic function is a polynomial function of degree 2. 2. It is generally written as $$f(x) = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants and $a \neq 0$.

1. The problem asks to write a quadratic function to the model. 2. A quadratic function generally has the form $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants.

1. The problem is to find a quadratic function $f(x)$ that models a parabola with vertex at $(0,4)$ and passing through points $(-2,8)$ and $(2,8)$. 2. Since the vertex is at $(0,4

1. The problem asks us to write a quadratic function modeling the given graph with x-intercepts at $x=2$ and $x=6$. 2. The factored form of a quadratic function with roots $r_1$ an

1. The problem is to solve the quadratic equation $$2x^2 - 4x - 6 = 0$$. 2. First, identify the coefficients: $$a = 2$$, $$b = -4$$, and $$c = -6$$.

1. **State the problem:** We need to find the missing number (let's call it $a$) in the equation $$4x - 10 = a x + 12$$ such that the equation has no solutions. 2. **Understand whe

1. **State the problem:** We are given that $y$ is directly proportional to $(x+3)^2$, and the sum of $y$ values when $x=8$ and $x=10$ is 870. 2. **Write the proportionality equati