🧮 algebra

1. **State the problem:** We are given matrices \(P = \begin{bmatrix}3 & 4 \\ 2 & x\end{bmatrix}\), \(Q = \begin{bmatrix}1 & 3 \\ -2 & 4\end{bmatrix}\), and \(R = \begin{bmatrix}-5

1. **State the problem:** We want to express the rational function $$\frac{x^2+x+4}{(1-x)(x^2+1)}$$ as a sum of partial fractions and check which of the given options (a), (b), (c)

1. Stating the problem: We are given the matrix $$P = \begin{bmatrix} y-2 & y-1 \\ y-4 & y+2 \end{bmatrix}$$ and its determinant $$|P| = -23$$. We need to find the value of $$y$$.

1. The problem asks which of the given numbers is not an integer. - An integer is a whole number without fractions or decimals.

1. **State the problem:** Find the range of values of $x$ for which $$x^2 + 4x + 5 < 3x^2 - x + 2.$$\n\n2. **Rewrite the inequality:** Move all terms to one side to compare with ze

1. The problem asks to simplify the product of three fractions: $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4}$. 2. Multiply the numerators together: $1 \times 1 \times 1 = 1$.

1. **State the problem:** We are given matrices \(P = \begin{bmatrix} 2 & 1 \\ 5 & -3 \end{bmatrix}\) and \(Q = \begin{bmatrix} 4 & -8 \\ 1 & -2 \end{bmatrix}\). We need to find \(

1. **State the problem:** Solve the inequality $$x^2 - 2x \geq 3$$ and analyze the given options. 2. **Rewrite the inequality:** Move all terms to one side to get zero on the other

1. The problem states that the roots of a quadratic equation are $-3$ and $1$. We need to find which of the given equations has these roots. 2. Recall that if a quadratic equation

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