🧮 algebra

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

1. **State the problem:** We need to subtract $-25$ from $-18$, which means calculating $-18 - (-25)$. 2. **Recall the subtraction rule:** Subtracting a negative number is the same

1. **State the problem:** Multiply the mixed numbers $1\frac{1}{5}$, $\frac{3}{4}$, and $2\frac{1}{2}$.\n\n2. **Convert mixed numbers to improper fractions:**\n- $1\frac{1}{5} = \f

1. The problem is to evaluate the expression $\frac{2}{0}$.\n\n2. Division by zero is undefined in mathematics because there is no number that you can multiply by 0 to get a non-ze

1. **Problem 4: Representing the triangle angles as a system of linear equations** The problem states:

1. **State the problem:** Solve the inequality $9x + 4 > 10$ for $x$. 2. **Recall the rule:** To solve inequalities, isolate the variable on one side. When dividing or multiplying

1. The problem states that the height $h$ of a person should not be less than 3.8 ft to ride the roller coaster. 2. This means the height must be at least 3.8 ft.

1. The problem states that Adam's average marks in subjects A, B, C, D, and E are more than 85. 2. To express the average, we sum the marks of all five subjects and divide by 5.

1. **State the problem:** We need to find the values of $p$ for which the inequality $$\frac{2}{3} - \frac{p}{4} \geq \frac{7}{6}$$ holds true. 2. **Write down the inequality:** $$

1. **Stating the problem:** We are given an arithmetic sequence with the formula for the $n$-th term: $$U_n = 12 - 5n$$

1. **State the problem:** We have three purchases with total costs and want to find the cost of one t-shirt ($t$), one video ($v$), and one stuffed animal ($s$).

1. **State the problem:** We need to find the number of adult ($x$), student ($y$), and child ($z$) tickets sold given:

1. **State the problem:** We need to draw the prime factor tree for 90 and then use it along with the prime factorization of 252 to find the highest common factor (HCF) of 90 and 2