🧮 algebra

1. **State the problem:** An employee earns 175 for 15 hours of work at a constant hourly rate. We need to find how much the employee will earn in 18 hours. 2. **Formula:** Earning

1. **State the problem:** Find the values of $c$ such that the angle between vectors $\mathbf{u} = (1,2,1)$ and $\mathbf{v} = (1,0,c)$ is $60^\circ$. 2. **Formula used:** The cosin

1. **State the problem:** We have two types of pumps, large and small, filling a swimming pool. Two large and one small pump fill the pool in 4 hours. One large and three small pum

1. The problem asks to identify which statements about rational and irrational numbers are true. 2. Recall definitions:

1. **State the problem:** Solve the inequality $x + 12 > 17$ for $x$. 2. **Write the inequality:**

1. **Problem statement:** 6. b) If the school only has one pump, how long will it take to fill the pool? Given answer: 9.6 hrs.

1. The problem asks to determine the coordinates of the vertex of the parabola shown on the graph. 2. The vertex of a parabola given by the equation $y = ax^2 + bx + c$ is found us

1. **Problem Statement:** Determine the coordinates of the vertex of the parabola from the graph. 2. **Understanding the Vertex:** The vertex of a parabola in vertex form is given

1. **State the problem:** Solve the equation $w(w+4) = -8$ using the quadratic formula and express the solution set in exact simplest form. 2. **Rewrite the equation:** Expand and

1. **State the problem:** Solve the quadratic equation using the quadratic formula and express the solution set in exact simplest form. 2. **Recall the quadratic formula:** For a q

1. **State the problem:** Solve the equation $2(w+4) = -8$ for $w$. 2. **Use the distributive property:** Multiply 2 by each term inside the parentheses.

1. **State the problem:** Solve for $x$ in the equation $$10^3 \cdot 10^x \cdot 10^2 = 10^5.$$\n\n2. **Recall the rule for multiplying powers with the same base:** When multiplying

1. **State the problem:** Solve the quadratic equation $$-2t^2 - 10t + 5 = 0$$ by completing the square and applying the square root property. 2. **Rewrite the equation:** First, d

1. **Problem statement:** We have a situation where $d$ represents the number of days of driving, and the expression $14 - 0.6d$ is given.

1. **State the problem:** Simplify the expression $6 - 2x + 5 + 4x$ and identify the error in Tyler's work. 2. **Tyler's error:** Tyler incorrectly grouped terms as $(6 - 2)x + (5

1. The problem is to find the constant term that completes the polynomial $m^2 - 4m + \_$ to make it a perfect-square quadratic. 2. A perfect-square quadratic has the form $\left(m

1. The user requests answers with 3 relevant numbers, which means answers should be rounded or expressed with three significant figures. 2. To demonstrate, let's solve a sample pro

1. The problem is to find the product of $6x$ and the polynomial $(x^2 - 4x - 3)$. 2. The formula used here is the distributive property: $a(b + c + d) = ab + ac + ad$.

1. **State the problem:** A laptop originally costs 209. The price is increased by 12%. We need to find the new price after the markup. 2. **Formula used:** To find the new price a

1. The problem asks to find the sale price of diapers originally costing 28.75 with a 10% markdown. 2. The formula for the sale price after a discount is: $$\text{Sale Price} = \te

1. **State the problem:** Solve the equation $$\frac{2tq}{5} = 0$$ for the variable $tq$. 2. **Recall the rule:** A fraction equals zero if and only if its numerator is zero (and t