🧮 algebra

1. Problem: If $(k - 3)$, $(2k + 1)$, and $(4k + 3)$ are three consecutive terms of an AP, find the value of $k$. Step 1: In an AP, the difference between consecutive terms is cons

1. **State the problem:** Solve the recurrence relation $$a_n + 7a_{n-1} + 8a_{n-2} = 0$$ for $$n \geq 2$$ with initial conditions $$a_0 = 2$$ and $$a_1 = -7$$. 2. **Find the chara

1. It seems you mentioned there is a wrong variable letter in a problem. 2. Please provide the exact problem or equation so I can identify and correct the variable letter.

1. Problem: If $(k - 3)$, $(2k + 1)$, and $(4k + 3)$ are three consecutive terms of an AP, find the value of $k$. Step 1: In an AP, the difference between consecutive terms is cons

1. **State the problem:** We have a two-digit number whose digits sum to 11.

1. **State the problem:** We have a circle with equation $$x^2 + y^2 - 4x + 8y - 8 = 0$$.

1. The problem is to solve the equation $$2x^2 - 4x - 6 = 0$$ for $x$. 2. First, simplify the equation by dividing all terms by 2 to make the coefficients smaller:

1. **State the problem:** We are given the quadratic equation in $x$: $$ (y + z)^2 x^2 + (y + z)(z + w) x + (z^2 + w^2 + 2zw) = 0 $$

1. The problem is to find the greatest common factor (GCF) of 16 and 24. 2. First, find the prime factorization of each number:

1. The problem asks to determine if the discriminant of the quadratic function $g(x)$ is positive, zero, or negative based on the graph description. 2. From the description, $g(x)$

1. Calculate $5/12 + 3/2 \times (-1/2)$. Multiply first: $3/2 \times (-1/2) = -3/4 = -9/12$.

1. **Problem a:** Solve the inequality $3x^2 + 5x - 2 \geq 0$. 2. Find the roots of the quadratic equation $3x^2 + 5x - 2 = 0$ using the quadratic formula:

1. The problem asks us to find all values of $k$ such that the graph of the quadratic function $$y = x^2 + 20x + k$$ does not intersect the x-axis. 2. The graph intersects the x-ax

1. The problem asks us to find all values of $k$ such that the graph of the quadratic equation $$y = x^2 + 6x + k$$ intersects the x-axis at two distinct points. 2. To find where t

1. The problem asks us to find all values of $k$ such that the quadratic equation $3x^2 - 18x + k = 0$ has real, unequal roots. 2. Recall that for a quadratic equation $ax^2 + bx +

1. **Problem 1:** Calculate $4^7 - 4^6$ and express $2^7 + 2^6$ as $a^b$ where $a,b$ are natural numbers. - Calculate $4^7 - 4^6$:

1. The problem asks us to find all values of $k$ such that the quadratic equation $kx^2 + 16x - 64 = 0$ has equal roots. 2. Recall that a quadratic equation $ax^2 + bx + c = 0$ has

1. **Problem statement:** A shopkeeper raises the price of a product by 50% on the cost price. We need to find the discount percentage on the marked price so that the shopkeeper st

1. **State the problem:** We need to find a two-digit number such that when we add 18 to the number formed by interchanging its digits, the sum of the digits of the original number

1. **State the problem:** We need to find how much more money is needed for Phase 3 to match the total cost of Phases 1 and 2 combined. 2. **Convert Phase 1 cost from base 4 to dec

1. **State the problem:** We need to find how much more money is needed for Phase 3 to match the total cost of Phases 1 and 2 combined. 2. **Convert Phase 1 cost from base 4 to dec