🧮 algebra

1. **State the problem:** Convert the point-slope form equation $$y + 3 = 5(x + 2)$$ into slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** Point-slope form is $$y -

1. **State the problem:** Convert the point-slope form equation $$y - 4 = -(x + 2)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the point-slope form:** $$y - y_1 = m(x - x

1. **Stating the problem:** We need to verify or solve the equation $\log_b a \cdot \log_c b \cdot \log_a c = 1$. 2. **Recall the change of base formula:** For any positive numbers

1. **State the problem:** Convert the point-slope form equation $$y + 2 = 5(x - 3)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** The point-slope form is $$y

1. **State the problem:** Convert the point-slope form equation $$y + 3 = 4(x - 5)$$ into slope-intercept form $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept.

1. **State the problem:** Convert the point-slope form equation $$y + 5 = -2(x - 1)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** The point-slope form is $$y

1. **Stating the problem:** We want to verify the identity $\log_a b \times \log_c a = \log_c b$. 2. **Recall the change of base formula:** For any positive numbers $x, y$ and base

1. **State the problem:** We start with the line $y = x$ and transform it into $y = \frac{x}{4} + 2$. We need to find the slope and describe how the line is shifted. 2. **Recall th

1. **Problem Statement:** Expand the expression $$\left(4x^3 + \frac{1}{2}x\right)^8$$ using the binomial theorem. 2. **Formula:** The binomial theorem states:

1. **Stating the problem:** Three friends share a sum of money in the ratio 2:3:5. One friend gets 1200. We need to find how much more one friend gets compared to another. 2. **Und

1. **Stating the problem:** We are given pairs of numbers connected by arrows: 90 \rightarrow 60, 30 \rightarrow 30, 18 \rightarrow 22, 12 \rightarrow 0, and a horizontal line with

1. **State the problem:** We are given the ratio of the present ages of two people, $a$ and $b$, as $3:4$. Five years ago, the ratio of their ages was $2:3$. We need to find their

1. **Stating the problem:** Solve the cubic equation $$x^3 + 3x^2 + 3x + 9 = 0$$ for the values of $x$. 2. **Recognize the form:** The equation resembles the expansion of a cube of

1. **Problem statement:** Three partners A, B, and C invest in a business in the ratio 4:5:6. The total profit is 120000, and A's share of the profit is 16000. We need to find the

1. **State the problem:** Solve the equation $$2x^4 + 6x^2 - 8 = 0$$ for $x$. 2. **Use substitution:** Let $y = x^2$. Then the equation becomes $$2y^2 + 6y - 8 = 0$$.

1. **State the problem:** Evaluate the function $f(x) = x^3 - 4x^2 - 11x + 30$ at $x = -3$. 2. **Recall the formula:** To find $f(-3)$, substitute $x$ with $-3$ in the function.

1. We are given the cubic function $$f(x) = x^3 + 4x^2 - 25x - 100$$ and one solution $$x = -5$$. We need to find the remaining solutions (roots) of the equation $$f(x) = 0$$. 2. S

1. **State the problem:** We need to divide the polynomial $$f(x) = x^3 - 2x^2 - 32x + 15$$ by the factor corresponding to the root $$x = -5$$. 2. **Recall the factor theorem:** If

1. **Problem statement:** (a) Factorise $x^2 - 16$

1. **Problem:** Simplify the surd expression $\sqrt{50}$. **Solution:** Use the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.

1. **Stating the problem:** Calculate the value of $7 \times 9 \times 3^{\frac{2}{9}}$. 2. **Formula and rules:** When you have a product involving an exponent, calculate the expon