🧮 algebra

1. **State the problem:** Factor the quadratic expression $$x^2 - 4x - 12$$. 2. **Recall the factoring formula:** For a quadratic in the form $$x^2 + bx + c$$, we look for two numb

1. **State the problem:** We need to find the equation of a parabola with vertex at $(1,-9)$ and roots (solutions) at $(-2,0)$ and $(4,0)$. 2. **Recall the vertex form of a parabol

1. **State the problem:** We want to divide the polynomial $$x^3 + 4x^2 + x - 6$$ by $$x - 2$$ using synthetic division. 2. **Recall the synthetic division setup:**

1. **State the problem:** We need to divide the polynomial $8x^6 - 6x^2 - 2x$ by the monomial $2x$. 2. **Formula and rule:** When dividing a polynomial by a monomial, divide each t

1. **State the problem:** We need to expand the product $$(x - 3)(x - 9)(x - 4)$$ and write it in the form $ax^3 + bx^2 + cx + d$. 2. **Use the distributive property (FOIL for bino

1. **State the problem:** Multiply the monomial $4x$ by the binomial $3x^3y^2 - 2xy^3$. 2. **Recall the distributive property:** To multiply a monomial by a binomial, multiply the

1. **State the problem:** Multiply the monomial $4x$ by the binomial $3x^3y^2 - 2xy^3$. 2. **Formula and rules:** To multiply a monomial by a binomial, use the distributive propert

1. **State the problem:** We need to add the two polynomials $$2x^3 + x^2 - 3$$ and $$2x^2 + 4x - 1$$. 2. **Recall the rule:** When adding polynomials, combine like terms. Like ter

1. **State the problem:** We need to order the terms of the polynomial $$2y^5 + y + 3y^7 - 4y^3 + 12$$ correctly. 2. **Recall the rule:** Polynomial terms are ordered by descending

1. **State the problem:** We need to find an equation for a cubic curve that passes through the points $(1,0)$ and $(2,1)$ and has an S-shape with an inflection near $(2,1)$. 2. **

1. **State the problem:** We need to find the equation of a parabola that opens downward, has a vertex at $(0,0)$, and passes through the point $(1,-2)$.

1. **State the problem:** We need to find the equation of a square-root function that passes through the points $(-1, -2)$ and $(0, -1)$. 2. **General form of square-root function:

1. The problem is to find an equation for a cubic-like S-curve passing through points $(1,0)$ and $(2,1)$. 2. A general cubic function has the form $$y = ax^3 + bx^2 + cx + d$$ whe

1. The problem is to find the equation of an absolute value graph with vertex at $(-3, -4)$ and passing through $(-2, -3)$. 2. The general form of an absolute value function with v

1. The problem asks us to identify the dependent variable in the equation $$m = 5 r + 4$$. 2. In an equation of the form $$y = mx + b$$, the dependent variable is the one that depe

1. **State the problem:** We need to find the price of a single movie ticket before tax, given that 15 tickets cost $147.66 including a 7% sales tax. 2. **Identify the formula:** T

1. **State the problem:** Solve the absolute value inequality $$|3x - 6| \leq 12$$. 2. **Recall the rule for absolute value inequalities:** For $$|A| \leq B$$ where $$B \geq 0$$, t

1. **State the problem:** We need to solve the absolute value equation $$|4x - 5| = 7$$ for $x$. 2. **Recall the rule for absolute value equations:** If $$|A| = B$$ where $B > 0$,

1. **State the problem:** We need to find the range of $x$ (number of copies) such that the value $V$ of the machine satisfies $$2000 \leq 3500 - 0.04x \leq 3000.$$ 2. **Write the

1. **State the problem:** Solve the inequality $$8x + 2 < 10x - 4$$. 2. **Isolate variable terms:** Subtract $$8x$$ from both sides:

1. **State the problem:** We need to write the ratio of acres to farms given that there are $3,500,000$ acres and $22,500$ farms. 2. **Write the ratio:** The ratio of acres to farm