🧮 algebra

1. **State the problem:** We need to find the possible values of the parameter $r$ such that the function $$h(x) = \log_{10}(4x^2 - rx + r - 1)$$ is defined for all real numbers $x

1. The problem is to solve the exponential equation $5^{x-2}=125$. 2. Recognize that 125 is a power of 5: $125 = 5^3$.

1. **State the problem:** Solve the system of equations: $$x + 3y = -5$$

1. We are given the problem: Solve the simultaneous equations (please provide the equations so I can help solve them). Since the equations were not specified, please provide them t

1. **State the problem:** We need to find the value of $x$ in the equation $3^{4x-1} = 81$. 2. **Rewrite the right side:** Note that $81$ can be expressed as a power of $3$ because

1. **State the problem:** We need to find the value of $k$ such that $$\frac{1}{3\sqrt{5}} - \frac{1}{2\sqrt{20}} = k\sqrt{5}.$$

1. We are given the system of equations: $$3x + y = 17$$

1. We are asked to simplify the expression $$\sqrt{12} - 6\sqrt{3} + \sqrt{3}$$. 2. First, simplify $$\sqrt{12}$$ by expressing 12 as $$4 \times 3$$ so:

1. Stated problem: Solve the equation $23 = 7x67$. 2. Interpret the equation: Here, $7x67$ means $7 \times 67$. So the equation is $23 = 7 \times 67$.

1. The problem is to solve the system of linear equations using the substitution method: $$2x + 4y = 4$$

1. **Stating the problem:** We need to solve the system of linear equations using the graphical method:

1. **State the problem:** We want to find the smallest number of emergency relief kits that each contain exactly one bottle of water, one can of food, and one beef jerky protein pa

1. Solve the equation $x^2 + 9x + 22 = 4$.\nStart by rewriting it as $x^2 + 9x + 18 = 0$.\nUse the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=9$, $c=

1. Solve the first equation $x^2 + 9x + 14 = 0$. Factor into $(x+7)(x+2)=0$.

1. Find the first four terms of the sequence. a) Since the sequence is unclear due to symbols, please clarify the exact form.

1. The problem asks us to find the solutions to a given quadratic equation. 2. The answers provided are $-2$ and $-4$.

1. The problem gives four quadratic equations and partly factored forms with one missing factor in each. We will complete the factoring and then find the two values of $x$ for each

1. Мавжуд ифодани кўриб чиқамиз: $$\sqrt{7} + 4\sqrt{3} + \sqrt{7} - 4\sqrt{3}$$. 2. $\sqrt{7}$ ва яна бир $\sqrt{7}$ қўшилади, демак уларнинг йиғинлари: $$\sqrt{7} + \sqrt{7} = 2\

1. State the problem: We want to find the mass of the wheat bag given that it is 15 kilograms more than the mass of the rice bag, which is 48 kilograms. 2. Define variables: Let $m

1. The problem is to simplify or understand the expression $3\sqrt{x}$. 2. The notation $3\sqrt{x}$ typically means $3 \times \sqrt{x}$, which is three times the square root of $x$

1. The problem is to simplify the expression involving a square root with 3 as a coefficient and the variable x inside the square root. 2. Start with the expression $$\sqrt{3x}$$.