🧮 algebra

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} x+2 & \text{if } x<0 \\ 2x & \text{if } x \geq 0 \end{cases}$$

1. The problem is to solve the equation or expression given by the user. However, since no specific problem was provided, I will demonstrate a general approach to solving algebraic

1. The problem is to factorize an algebraic expression. Since no specific expression was given, let's consider a general approach. 2. Factorization involves expressing a polynomial

1. **State the problem:** Solve the quadratic equation $-17x + x^2 = -12$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **Problem Statement:** The company produces two gadgets, x and y. We have two equations:

1. **State the problem:** Solve for $x$ in the equation $\log_3 2 + 2 \log_3 x = \log_3 (7x - 3)$. 2. **Recall logarithm properties:**

1. The problem involves direct variation, where $y$ varies directly as $x$. This means the relationship can be expressed as $y = kx$, where $k$ is the constant of variation. 2. To

1. The problem is to determine whether the number 401.16666667 rounds to 401.26 or 401.25 when rounded to two decimal places. 2. The rule for rounding to two decimal places is to l

1. **Problem statement:** A train travels 200 km at a uniform speed. If the speed had been 10 km/hr less, the journey would have taken 40 minutes (which is $\frac{2}{3}$ hours) mor

1. **State the problem:** We want to find values of $a$ and $b$ such that the piecewise function $$

1. The problem is to find the Least Common Multiple (LCM) of given numbers. 2. The LCM of two or more integers is the smallest positive integer that is divisible by each of the num

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle that to isolate $x$, we perform inverse operations to both sides of the equat

1. **State the problem:** We are given the function $$f(x) = 2^x \log_3 \left(7^{x^2 - 4}\right)$$ and asked to analyze it. 2. **Recall the logarithm power rule:** $$\log_a b^c = c

1. **Problem Statement:** We are given multiple equations and functions to analyze, including circles, points, and polynomial functions. We will solve and explain each part step-by

1. समस्या को समझें: मई से जून 2004 तक कारों के पंजीकरण में प्रतिशत वृद्धि ज्ञात करनी है। 2. दिए गए डेटा के अनुसार, मई में कारों के पंजीकरण = 17 हजार और जून में = 28 हजार।

1. The problem asks us to determine which car generated the lowest income for the company over the whole year and to find out how much that income was. 2. To solve this, we need to

1. **Problem statement:** We need to find in which quarter the car-sales company's total income was the lowest and how much it was. 2. **Given data:** The total incomes for each qu

1. **Problem statement:** Solve the system of linear equations: $$2x + 5y = 32$$

1. **Problem statement:** Find all real values of $\lambda$ such that the quadratic equation $$(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$$ has exactly one root in the interval $(0,1)

1. Let's start by understanding the problem in question 98. Since the exact problem isn't provided, I'll explain a common approach to solving algebraic equations, which is often th

1. **Problem statement:** Find all real values of $\lambda$ such that the quadratic equation $$(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0$$ has exactly one root in the interval $(0,1)