🧮 algebra

1. Stating the problem: Calculate the sum of all natural numbers:

1. **Problem 1:** Blake uses 60% of his paycheck to pay bills and has 320 remaining. Find the total paycheck amount. 2. **Formula:** Let $x$ be the total paycheck. Since Blake uses

1. **State the problem:** We know 36 men can slash a field of 160 meters in 10 days. We want to find how long it will take 48 men to slash a field of 128 meters.

1. Statement of the problem: Solve the quadratic equation $7x^2+23x=60$.\n 2. Write the equation in standard form by moving all terms to one side: $$7x^2+23x-60=0$$.\n

1. **Problem 7:** Solve the system of inequalities: $$\begin{cases} 4x - 2y < 4 \\ y \leq \frac{3}{4}x + 3 \end{cases}$$

1. **Problem (a): Factorise fully** $6y^2 - 5y - 4$ 2. To factorise a quadratic expression of the form $ay^2 + by + c$, we look for two numbers that multiply to $a \times c$ and ad

1. **State the problem:** Simplify the expression $$\frac{\sqrt{63} - \sqrt{7}}{\sqrt{28} - \sqrt{7}}$$. 2. **Recall the formula and rules:** To simplify expressions with square ro

1. **State the problem:** Emma bought 4 identical notebooks and a pen for 3 each, and her total was 35. We need to find the cost of each notebook. 2. **Define variables:** Let the

1. **Problem statement:** We need to find the equation of the locus of points equidistant from two points $P(2,5)$ and $Q(8,3)$. This locus is the perpendicular bisector of the seg

1. The problem is to solve the equation or expression where the solution is not included in the given answers. 2. First, identify the equation or expression clearly and write it do

1. **State the problem:** We need to find the cubic function $f(x) = ax^3 + bx^2 + cx + d$ that has a local maximum at $(2,4)$ and an inflection point at $(1,2)$. 2. **Recall the c

1. The problem asks for the absolute maximum value of the function $f(x) = 5$ on the interval $[2,10]$. 2. The function $f(x) = 5$ is a constant function, meaning it has the same v

1. **State the problem:** We need to determine which coupon, the 40% off or the 10 off, gives the lower price for each item. 2. **Formula for each coupon:**

1. **State the problem:** We need to determine which coupon gives the lower price for each item: a 40% off coupon or a 10 off coupon. 2. **Formula for each coupon:**

1. The problem asks to find the interval where the function $f(x) = 3|x + 1| - 2$ is increasing. 2. Recall that the absolute value function $|x|$ is defined as:

1. **State the problem:** We need to find the interval where the function $f(x) = -|x| + 1$ is decreasing. 2. **Recall the function and its properties:** The function is defined as

1. **Problem statement:** We are given the function $f(x) = x^3 + 3x^2 + kx - 1$ and need to find the range of values for $k$ such that $f$ is one-to-one. 2. **Key concept:** A fun

1. **Problem Statement:** We have four items with given prices and two coupons: a 40% off coupon and a 10 off coupon. We need to find which coupon gives the lower price for each it

1. **Problem Statement:** We are given the function $$f(x) = \frac{1}{3}x^3 + (k+1)x^2 + (k^2+5)x$$ and asked to analyze it for different values of the parameter $$k$$ based on the

1. Let's practice solving a quadratic equation: Solve for $x$ in $$x^2 - 5x + 6 = 0$$. 2. The formula to solve quadratic equations is the quadratic formula: $$x = \frac{-b \pm \sqr

1. **State the problem:** Solve the system of simultaneous equations: $$6x + 5y = -6$$