🧮 algebra

1. **Problem:** Find the square root of the discriminant of the quadratic equation $$2x^2 + 3x - 5 = 0$$. 2. The discriminant $$\Delta$$ of a quadratic equation $$ax^2 + bx + c = 0

1. The problem asks for the sum of the roots of the quadratic equation $$3x^2 - 5x + 2 = 0$$. 2. Recall that for a quadratic equation $$ax^2 + bx + c = 0$$, the sum of the roots is

1. **State the problem:** We have populations of four countries: England, Wales, Scotland, and Northern Ireland, given in scientific notation and ordinary numbers. We want to under

1. **State the problem:** Mrs. Allen made a deposit of $150 and 11 monthly payments of $118 each to buy a refrigerator. 2. **Calculate the total amount of the monthly payments:**

1. Stating the problem: Simplify the expressions \(\frac{2a - 1}{5a} - \frac{4b - 3}{10b}\) and \(\frac{2x + 1}{x} + \frac{3y - 2}{y} - 5\). 2. Simplify the first expression:

1. The problem is to factor the expression $x^2$. 2. Notice that $x^2$ is a perfect square, which can be written as $x \times x$.

1. **State the problem:** We have a car with a sale price of 15200.

1. We are given the expression $\sqrt[3]{\frac{81}{3}}$ and need to write it in the form $3^k$ and find $k$. 2. Simplify the fraction inside the cube root:

1. **Simplify** $\frac{5y^4}{y^{-2}}$. Step 1: Recall the rule for dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$.

1. **State the problem:** We are given the ratio of analysts to field agents as 3:5 and the total number of agents as 40. We need to find the number of analysts. 2. **Understand th

1. The problem asks for the domain of the function defined by the absolute value expression $|x^3 - 7|$. 2. The domain of a function is the set of all input values $x$ for which th

1. **State the problem:** We are given a stove with a cash price of 1500.

1. **State the problem:** We have a rectangle with an original length of 8 cm and an area of 20 cm². We want to find the original width, then analyze what happens when the length i

1. **State the problem:** We have a stove with a cash price of 800.

1. Let's clarify the problem: You mentioned the answer to the first question was 6, 8, 10. 2. These numbers (6, 8, 10) form a Pythagorean triple, which means they satisfy the Pytha

1. The problem is to find the pattern or rule for the sequence 6, 8, 10. 2. Observe the differences between consecutive terms: $8 - 6 = 2$ and $10 - 8 = 2$.

1. The problem asks for the multiplication inverse of the fraction $\frac{a}{b}$. 2. The multiplication inverse (also called the reciprocal) of a number is the number which, when m

1. The problem asks to find 33 and a third percent of 100. 2. First, convert 33 and a third percent to a fraction or decimal. 33 and a third percent is equal to $33\frac{1}{3}\% =

1. **State the problem:** We have a club with $x$ adults. Out of these, some are over 30 years old.

1. **Calculate the sum and difference of the given numbers:** Given expression:

1. **Problem statement:** Given positive numbers $a, b, c$ such that $a + b + c = 18$, prove that $$\frac{a}{b^2 + 36} + \frac{b}{c^2 + 36} + \frac{c}{a^2 + 36} \geq \frac{1}{4}.$$