🧮 algebra

1. **State the problem:** We are given the function $f(x) = 3x$ and want to understand its properties. 2. **Formula and rules:** This is a linear function of the form $f(x) = mx +

1. The problem is to find the number that must be added to the equation $x^2 - 4x = 4$ to complete the square. 2. The method of completing the square involves creating a perfect sq

1. **State the problem:** Factor the expression $x^4 - 16$. 2. **Recall the formula:** This is a difference of squares since $x^4 = (x^2)^2$ and $16 = 4^2$.

1. The problem is to find the number that must be added to the equation $x^2 - 4x = 4$ to complete the square. 2. The method of completing the square involves creating a perfect sq

1. **State the problem:** Factor the polynomial $$x^3 + x^2 + x + 1$$. 2. **Identify the method:** We can try factoring by grouping since the polynomial has four terms.

1. **State the problem:** Factor the expression $x^2$. 2. **Recall the factoring rules:**

1. **State the problem:** Factor the quadratic expression $x^2 - 4x + 3$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. The problem is to explain how the vertical and horizontal asymptotes of the function $$f(x) = \frac{22x - 3}{x + 13}$$ are found. 2. **Vertical asymptote:** This occurs where th

1. The problem asks for the percentage of 1000 in a million. 2. The formula to find the percentage is: $$\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 10

1. The problem asks: What is 1% of 1 million? 2. To find a percentage of a number, use the formula:

1. **State the problem:** We are given three lines forming a triangle with equations $x+2y=4$, $2x-y=3$, and $x-y-2=0$. We need to prove the triangle is right angled and find its a

1. **State the problem:** We are given three lines forming a triangle with equations $x+2y=4$, $2x-y=3$, and $x-y-2=0$. We need to prove the triangle is right angled and find its a

1. **Problem statement:** A movie theatre's profit $P$ depends on the number of $1 price increases $x$ as $$P = 20(15 - x)(11 + x).$$

1. **State the problem:** Solve the inequalities: a) $\frac{x}{2} + 1 < 3$

1. **Problem:** Find the unknown value □ in the proportion $3 : 4 = 6 : \square$. 2. **Understanding Proportions:** A proportion states that two ratios are equal. The formula for p

1. **State the problem:** We are given the function $$y = (x^2 + x^{-2})^2$$ and want to understand its shape and properties. 2. **Rewrite the function:** Expand the square using t

1. **State the problem:** We are given the function $$y = (x^2 + x^{-2})^2$$ and want to understand its behavior and graph. 2. **Recall the formula and rules:** The function involv

1. The problem states: "The product of b and 155, added to 359 is equal to 397." We need to write this as an equation. 2. "The product of b and 155" means multiply b by 155, which

1. The problem asks us to write a quadratic equation in the form $ax^2 + bx + c = 0$ with roots 6 and 7. 2. Recall that if a quadratic equation has roots $r_1$ and $r_2$, it can be

1. **State the problem:** We need to write a quadratic equation in the form $ax^2 + bx + c = 0$ whose roots are 6 and 7. 2. **Recall the formula:** If a quadratic equation has root

1. **Problem:** Solve the quadratic equation $x^2 + 6x - 10 = 0$ using the completing the square method. 2. **Formula and rules:** To complete the square for $x^2 + bx + c = 0$, re