🧮 algebra

1. Problem: Factor out the common binomial factor $(x+2)$ from an expression. 2. Suppose the expression is of the form $(x+2)(ax+b) + (x+2)(cx+d)$. Our goal is to factor $(x+2)$ ou

1. **State the problem:** We are given the function $f(x) = |x|$ and need to find its range. 2. **Recall the definition of the absolute value:** The absolute value $|x|$ of a numbe

1. The problem gives two functions: a) $f(x) = |x|$

1. \nWe are given three points: $(-3, a)$, $(0, 4)$, and $(6, 2a)$. We need to find $2a + 4$ given that these points are collinear.\n\n2. \nIf three points are collinear, the slope

1. **Énoncé du problème :** Étudier la fonction quadratique $$f(x) = (m+2)x^2 - 2(m+1)x + m - 1$$ avec $m \neq -2$. On demande de discuter les solutions de l'équation $f(x)=0$, ana

1. Stating the problem: Solve the equation $$2^{\frac{1}{2x}} + 2^{\frac{1}{x}} = 6$$ for $x$. 2. Introduce a substitution: Let $$y = 2^{\frac{1}{2x}}$$. Then, $$2^{\frac{1}{x}} =

1. **State the problem:** Solve the equation $$\frac{1}{2x} + \frac{1}{2} = 6$$ for $x$. 2. **Isolate the fraction terms:** Write the equation clearly:

1. **State the problem:** We need to simplify the function $f(x) = \frac{x^2 - 9}{x - 3}$ and identify which linear function among the options A, B, C, and D represents $f(x)$. 2.

1. The problem involves a function $f(x)$ and the expression $f(4) - f(-4) = f(b)$. We are asked to find the value(s) of $b$ such that this equality holds. 2. To solve this, we obs

1. **State the problem:** We have a geometric sequence with common ratio $r$. The difference between the 2nd term $a r$ and the 5th term $a r^4$ is 156.

1. **State the problem:** We are given that $x$ and $y$ are positive real numbers and satisfy the system:\n$$\log_5(x+y) + \log_5(x-y) = 3$$\nand\n$$\log_2 y - \log_2 x = 1 - \log_

1. State the problem: Calculate the value of $$M = \left(-\frac{3}{8} \times -\frac{4}{7}\right) \div \left(\frac{3}{5} - \frac{5}{3} + 27 \frac{39}{19}\right)$$. 2. Calculate the

1. Stated problem: Determine if $\frac{-12}{8} = -1.5$ is correct. 2. Perform the division: $\frac{-12}{8} = -1.5$ because dividing -12 by 8 gives -1.5.

1. **Stating the problem:** Solve the equation $$4^{-x+\frac{1}{2}} - 7 \cdot 2^{-x} - 4 = 0$$. 2. **Expressing powers with the same base:** Note that $$4 = 2^2$$, so $$4^{-x+\frac

1. The problem is to evaluate the expression $$-\frac{12}{4} \times \frac{8}{4} = ?$$ 2. First, simplify each division:

1. Start with the original expression: $-12 + \sqrt{144} - 144$. 2. Calculate the square root: $\sqrt{144} = 12$.

1. State the problem: Solve the quadratic equation $3x^2+2x-8=0$ using the factorizing method. 2. Multiply the coefficient of $x^2$ (which is 3) by the constant term (which is -8):

1. The problem asks for the difference in cost between 300 g of potatoes and 300 g of carrots. 2. From the graph, the cost of 300 g of potatoes is approximately $27$ p.

1. The problem is to solve the equation $n^2 = 7^n$ for $n$ using the Lambert W function. 2. Rewrite the equation: $n^2 = 7^n$ implies $n^2 = e^{n \ln 7}$.

1. **State the problem:** We need to find the value(s) of $n$ such that $$n^2 = 7^n$$. 2. **Analyze the equation:** The equation is $$n^2 = 7^n$$. Here, $n^2$ is a quadratic functi

1. Problem: Simplify the expression $\sqrt[3]{4^{1-5}}$. 2. Compute the exponent inside the power: $1-5=-4$.