🧮 algebra

1. **State the problem:** Given the quadratic function $$h = -0.2 d^2 + 0.8 d + 2,$$ find the value of $$h$$ at $$d = -1,$$ and determine the vertex of the parabola which represent

1. The problem is to convert the mixed number 6 5/9 to an improper fraction. 2. Recall that a mixed number consists of a whole number and a fraction: here, 6 and 5/9.

1. The problem is to convert the improper fraction $\frac{43}{7}$ into a mixed number. 2. Divide the numerator (43) by the denominator (7) to find the whole number part: $43 \div 7

1. **Statement:** Simplify the expression $$\sqrt{6} + \sqrt{\frac{17}{3}} + \frac{\sqrt{98} + \sqrt{338}}{\sqrt{72}}$$. 2. **Simplify each radical inside the expression:**

1. State the problem: Solve the quadratic equation $$2x^2-8x+8=0$$ using the complete the square method. 2. Divide the entire equation by 2 to simplify: $$x^2 - 4x + 4 = 0$$.

1. The problem asks to evaluate the sum $a = \sqrt{\frac{19}{2,(1)}} + \sqrt{\frac{20}{2,(2)}} + \sqrt{\frac{21}{2,(3)}} + \dots + \sqrt{\frac{26}{2,(8)}}$. 2. We need to clarify t

1. **State the problem:** We are given a geometric series: $100 + 90 + 81 + \ldots$ and need to show that the sum of the first $n$ terms is given by $$ \text{sum} = 1000 \left(1 -

1. **Énoncé du problème :** Soit un groupe $G$ avec la loi multiplicative, tel que pour tout $g \in G$, $g^2 = e$ où $e$ est l'élément neutre. Montrer que $G$ est abélien.

1. **Simplify the expression $\sqrt{26} - 1$.** Since $\sqrt{26}$ is approximately 5.099, the expression is approximately

1. **State the problem:** We want to describe and understand the region defined by the inequalities: $$-1 < x < 4$$

1. We are asked to add the two fractions $\frac{4}{10}$ and $\frac{52}{60}$. 2. To add fractions, we first find a common denominator. The denominators are 10 and 60.

1. The problem is to add the fractions $\frac{4}{10}$ and $5\ \frac{2}{20}$.\n2. Convert the mixed number $5\ \frac{2}{20}$ to an improper fraction: $5 = \frac{5 \times 20}{20} = \

1. We are asked to find the non-real solutions to the quadratic equation $x^2 + 4x + 5 = 0$ using the quadratic formula. 2. Recall the quadratic formula for solutions of $ax^2 + bx

1. The topic is analyzing algebraic functions, which involves understanding properties like intercepts, extrema, domain, and range. 2. A beginner-level question could be: Analyze t

1. **Problem statement:** Given sets $A = B = \{x \mid -2 \leq x \leq 2\}$ and functions: - (a) $f(x) = |x|$

1. Stating the problem: Solve the equation $$\frac{14}{2^x + 3} + \frac{15}{2^x + 1} = 5$$ for $x$. 2. Let $y = 2^x$. Since $2^x > 0$ for all real $x$, we have $y > 0$.

1. **State the problem:** Solve for $x$ in the equation $$\frac{14}{2^x}+3 + \frac{15}{2^x} + 1 = 5.$$\n\n2. **Combine like terms:** Group terms with $2^x$: $$\frac{14}{2^x} + \fra

1. **Énoncé du problème** : Démontrer que la relation $R$ sur $G$ définie par $xRy \iff x = y$ ou $x = y^{-1}$ est une relation d'équivalence.

1. We start with the equation: $$16^{1/x} - 20 \cdot 2^{(2/x)-2} + 4 = 0$$ 2. Rewrite 16 as a power of 2: $$16 = 2^4$$, so $$16^{1/x} = (2^4)^{1/x} = 2^{4/x}$$.

1. **State the problem:** Solve the system of linear equations by matrix methods for two systems: ①\

1. **Problem:** Suppose $R = \{(x,y): y \geq x^2 - 4 \text{ and } y < x + 2\}$. Find the domain and range of $R^{-1}$. **Step 1:** The relation $R$ consists of points $(x,y)$ where