🧮 algebra

1. **State the problem:** We want to estimate the solution to the equation $$x^3 + 7x - 5 = 0$$ between $$x=0$$ and $$x=1$$ using the iteration formula $$x_{n+1} = \frac{5}{x_n^2 +

1. **State the problem:** Show that the equation $$x^3 + 7x - 5 = 0$$ has a solution between $$x=0$$ and $$x=1$$. 2. **Evaluate the function at $$x=0$$:**

1. **State the problem:** Write the quadratic expression $x^2 + 6x - 7$ in the form $(x + a)^2 + b$ where $a$ and $b$ are integers. 2. **Recall the formula:** To complete the squar

1. **Problem Statement:** Given a circle with equation $$x^2 + y^2 = 25$$ and a line $$3x + 4y + 1 = 0$$, find:

1. **State the problem:** Yasmin, Kian, and Maggie have scores in the ratio 3 : 5 : 12, and the mean of their scores is 180. We need to find Yasmin's score. 2. **Understand the rat

1. **State the problem:** We are given a table of values for $t$ (days) and $D(t)$, and we want to find which function among the given options models the data correctly. 2. **Given

1. **State the problem:** Solve the equation $$\left(\frac{2}{3}\right)^{x+1} = \left(\frac{16}{9}\right)^{x-5}$$ for $x$. 2. **Rewrite the bases:** Note that $$\frac{16}{9} = \lef

1. **State the problem:** Simplify the expression $$\left(\frac{750}{512}\right)^{\frac{1}{3}}$$ which means finding the cube root of the fraction $\frac{750}{512}$. 2. **Recall th

1. **State the problem:** Simplify the expression $$\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$$. 2. **Recall the rule:** Dividing by a fraction is the same as multiplying

1. The problem is to simplify the expression $\frac{a}{a-b} \div \frac{a}{a-c} \div \frac{a}{a-d}$. 2. Recall that dividing by a fraction is the same as multiplying by its reciproc

1. The problem asks for the range of the function represented by the graph. 2. The graph is a parabola opening upwards with its vertex at approximately $(0,1)$.

1. **State the problem:** Solve the equation $x = 2y + x^4$ for $y$ in terms of $x$. 2. **Rewrite the equation:** We want to isolate $y$ on one side. Starting with:

1. The problem gives two functions: $$f(x) = x^2$$ and $$g(x) = (x - 9)^2 + 3$$. 2. We are asked to understand or analyze these functions. Both are quadratic functions, which means

1. The problem is to rewrite the quadratic function $$f(x) = x^{2} + 16x - 46$$ by completing the square. 2. The formula for completing the square for a quadratic $$ax^{2} + bx + c

1. **State the problem:** Simplify the expression $$3(n + 3mn) - 5n(2m - 1)$$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the factor outside

1. **Problem a:** Simplify and solve \(\left(2 \frac{1}{3} + \frac{1}{2} - \frac{1}{12}\right) - 2 \left(\frac{1}{3} + \frac{1}{4}\right) - \sqrt{\frac{1}{9}}\). 2. Convert mixed n

1. **Problem 9a:** Predict the next three numbers in the pattern: $-1, \frac{1}{2}, -\frac{7}{8}, -\frac{1}{4}, \frac{3}{8}, ...$ - Observe the pattern: signs alternate and fractio

1. **Problem:** Lori owes her mother 39. She pays back 1/3 of this debt, then 1/4 of the remaining debt. How much does she still owe? 2. **Formula and rules:** To find the remainin

1. Calculate the following sums and differences of rational numbers. 1.a) $\frac{3}{10} + \frac{1}{5}$

1. Calculate $\frac{3}{10} + \frac{1}{5}$. Step 1: State the problem.

1. **State the problem:** Solve the equation $y = 2x + \frac{14}{15}$ for $y$ in terms of $x$. 2. **Understand the equation:** This is a linear equation where $y$ depends on $x$.