🧮 algebra

1. The problem states that the roots of the quadratic equation $$2x^2 - 5x + m = 0$$ are $$\alpha$$ and $$\beta$$, and we need to find $$\alpha^2 + \beta^2$$ in terms of $$m$$. 2.

1. The problem is to expand the expression $ (m+5)(m-2) $. 2. Use the distributive property (FOIL method) to multiply each term in the first parenthesis by each term in the second

1. **State the problem:** Factorise the expression $g^2 + 7g$. 2. **Identify common factors:** Both terms $g^2$ and $7g$ have a common factor of $g$.

1. The problem states that \(\alpha\) and \(\beta\) are roots of a quadratic equation with \(\alpha + \beta = 3\) and \(\alpha \beta = 2\). 2. Recall that for a quadratic equation

1. The problem asks to evaluate $\log_{0.25} 8$ and match it with one of the given options: (a) $\frac{3}{2}$, (b) $\frac{2}{4}$, (c) $\frac{2}{3}$, (d) $-\frac{3}{2}$.\n\n2. Recal

1. The problem states that the determinant of the matrix \(\begin{bmatrix}4 & x \\ 5 & 3\end{bmatrix}\) is 32. 2. Recall that the determinant of a 2x2 matrix \(\begin{bmatrix}a & b

1. The problem is to find the missing number in the series: 64, 16, 32, 8, ?.\n\n2. Observe the pattern between the numbers:\n- From 64 to 16: divide by 4 ($64 \div 4 = 16$)\n- Fro

1. **State the problem:** We need to find the value of the expression $$\frac{3x^2 + \frac{7}{y^2} - 8 - x^3(z^2 - 1)}{y^3 + 13}$$ given $x = -2$, $y = -3$, and $z = 2$. 2. **Calcu

1. **State the problem:** Rebecca must give 5 out of every 6 dollars she earns to the tax department. She keeps the remaining amount. This week, she kept 10 dollars. We need to fin

1. نبدأ بكتابة الدالة المعطاة: $$ د(س) = س + 3 \times س - 5 $$ 2. نطبق عملية الضرب: $$ 3 \times س = 3س $$

1. **State the problem:** Solve the system of equations: $$\frac{3}{5}a + \frac{1}{3}b = 3$$

1. **State the problem:** We need to find the value of $q$ in the expression $$(\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}.$$ 2. **Expand the product:** Use the dis

1. **State the problem:** Solve the equation $$\frac{1}{2}(x+2) - \frac{1}{4}(x-4) = \frac{1}{8}(x+4)$$ for $x$. 2. **Distribute the fractions:**

1. The problem is to understand what the natural logarithm function \(\ln\) means and why \(\ln(1)=0\), \(\ln(e)=1\), and \(\ln(e^2)=2\). 2. The natural logarithm \(\ln(x)\) is the

1. The problem is to simplify the expression by factoring out the common multiplier: $$500 \times 79 - 250 \times 157$$. 2. Identify the common multiplier between 500 and 250. The

1. **Aufgabe 2: Potenzen von 2 und -5** Gegeben sind Potenzen von 2 und -5 mit verschiedenen Exponenten. Wir berechnen jede Potenz einzeln.

1. **State the problem:** Find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of the polynomials $16a^4 - 4a^2 - 4a - 1$ and $8a^3 - 1$. 2. **Factorize the first p

1. **State the problem:** Prove that 6 is a factor of $n(n^2 + 5)$ for all integers $n \geq 1$ using mathematical induction. 2. **Base case:** For $n=1$, evaluate $1(1^2 + 5) = 1 \

1. **State the problem:** We need to prove that 6 is a factor of the expression $n(n^2 + 5)$ for all integers $n \geq 1$. 2. **Rewrite the expression:** The expression is $n(n^2 +

1. **State the problem:** A man's salary in 1975 was 2000 Naira per year, and it increased by 10% each year. We need to find the total amount he earned from 1975 to 1984 inclusive.

1. The problem is to find the square root of 13. 2. The square root of a number $x$ is a value $y$ such that $y^2 = x$.