🧮 algebra

1. **State the problem:** Solve the quadratic equation $$2x^2 + 9x - 5 = 0$$ for $x$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where

1. The problem is to find the value of $9^{\frac{3}{4}}$. 2. Recall the rule for fractional exponents: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$.

1. **State the problem:** Solve the equation $$\frac{x}{x-1} + \frac{x}{x+1} - \frac{3}{x-1} = 4$$ using the quadratic formula. 2. **Rewrite the equation:** Move all terms to one s

1. **State the problem:** Evaluate the expression $1^3 + 4 \cdot 1^2 - 11 \cdot 1 - 30$. 2. **Recall the order of operations:** Calculate exponents first, then multiplication, and

1. **State the problem:** Solve the equation $$\Delta an(2x - \pi) = \frac{\sqrt{2}}{5}$$ for $$x$$. 2. **Recall the definition and properties:** The function $$\Delta an$$ is the

1. **State the problem:** Factor the quadratic expression $7x^2 + 14x - 105$ and find the values of $x$ that satisfy $7x^2 + 14x - 105 = 0$. 2. **Write the equation:**

1. El problema nos pide escribir una ecuación que represente el valor de un iPad que cuesta 700 y se deprecia un 20% cada año. 2. La fórmula para depreciación exponencial es:

1. We are asked to simplify the expression $$\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2 + \sqrt{3}}}$$. 2. To simplify a fraction with square roots in numerator and denominator, we can mul

1. **State the problem:** Simplify and analyze the quadratic expression $-4x^2 - 32x - 28$. 2. **Formula and rules:** To simplify, factor out the greatest common factor (GCF) from

1. The problem asks for the end behavior of the polynomial function $$f(x) = -7x^6 - 63x^5 - 77x^4 + 343x^3 + 168x^2 - 700x + 336$$. 2. To determine end behavior, focus on the term

1. **State the problem:** We need to find the end behavior of the polynomial function $$f(x) = 3x^4 - 63x^2 + 6x^3 + 120 - 66x$$. 2. **Recall the rule for end behavior of polynomia

1. **State the problem:** Graph the quadratic function $g(x) = -3x^2$, plot points, sketch the curve, and find domain and range. 2. **Make a table of values:** Choose $x$ values: $

1. The problem is to understand why subtracting $\pi$ from 2 is done or what it means. 2. Subtraction is a basic arithmetic operation where you take away one number from another.

1. **Problema:** Semplifica l'espressione $2x(x^2+1)-3x(x^2-x+1)-(-x^3+x^2-x)$. 2. **Formula e regole:** Per semplificare, applichiamo la proprietà distributiva $a(b+c) = ab + ac$

1. **State the problem:** Solve the equation $$(x + 5)^2 - 16 = 0$$ for $x$. 2. **Use the formula:** Recognize this as a difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ where

1. **State the problem:** Find the magnitude of the vector $\vec{v} = (-3, -10)$. The magnitude of a vector $\vec{v} = (x, y)$ is given by the formula: $$|\vec{v}| = \sqrt{x^2 + y^

1. **State the problem:** We have a city population of 89,000 in 2010.

1. **State the problem:** We want to find for which values of $b$ the quadratic expression $$x^2 + bx + 12$$ is factorable over the integers. 2. **Recall the factorization rule:**

1. **State the problem:** We are given the equation $$(a \times 10^b) \times (5.3 \times 10^8) = 2.226 \times 10^4$$ where $a \times 10^b$ is in standard form. We need to find the

1. **State the problem:** Calculate $\left(5.46 \times 10^{-4}\right)^3 + \left(2.23 \times 10^{-5}\right) \times \left(1.39 \times 10^{-6}\right)$ and express the answer in standa

1. **State the problem:** We are given two quadratic functions: - $g(x) = 3x^2$