🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{3a^5+a^4-3a^3-3a^2+2}{1-a^2}$$. 2. **Factor the denominator:** Note that $$1 - a^2$$ is a difference of squares, which fac

1. The problem is to find the value of $A+B$ given variables $A$ and $B$. 2. Without specific values or expressions for $A$ and $B$, we cannot numerically evaluate $A+B$.

1. **State the problem:** Convert the postfix expression $$A B * C D ^ / E F G + * + H I + J K - / -$$ to its equivalent infix form, then evaluate it using values $$A=100, B=10, C=

1. **Problem 8:** The cost of one eraser and three pencils is 17, and the cost of three erasers and four pencils is 31. Find the cost of 5 erasers. 2. Let the cost of one eraser be

1. The problem is to rewrite the quadratic function $y = x^2 + 4x - 12$ into vertex form. 2. Recall that the vertex form of a quadratic is $y = a(x-h)^2 + k$, where $(h,k)$ is the

1. Stating the problem: Solve the quadratic equation $2x^2 + 3x - 4 = 0$ using the quadratic formula. 2. The quadratic formula to find roots of $ax^2 + bx + c = 0$ is given by:

1. The problem is to simplify the expression $\sqrt{28} + \sqrt{63}$. 2. First, factor the numbers under the square roots to find perfect squares.

1. The problem is to simplify the expression $\sqrt{1000} + \sqrt{90}$. 2. Simplify $\sqrt{1000}$ by factoring out perfect squares:

1. The problem is to simplify the expression $\sqrt{300} - \sqrt{48}$.\n2. Start by simplifying each square root. Break down $300$ and $48$ into their prime factors:\n $$300 = 10

1. We are asked to simplify the expression $\sqrt{8} + \sqrt{2} + \sqrt{72}$.\n2. Start by simplifying each square root where possible:\n - $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt

1. Stating the problem: Simplify the expression $\sqrt{200} - \sqrt{32}$.\n\n2. Simplify each surd by factoring out the perfect squares:\n\n$\sqrt{200} = \sqrt{100 \times 2} = \sqr

1. Stating the problem: Solve the quadratic equation $$3x^2 + x - 2 = 0$$ using the method of completing the square. 2. First, divide the entire equation by 3 to make the coefficie

1. The problem asks to evaluate the expression $$\frac{\sqrt[3]{5} - \sqrt{3}}{\sqrt[3]{5} - \sqrt{3}}$$. 2. Notice that the numerator and denominator are identical.

1. **Problem:** When loaded with bricks, a lorry has a mass of 11,600 kg. The mass of the bricks is three times that of the empty lorry. Find the mass of the bricks. **Step 1:** Le

1. We are given the quadratic equation $3x^2 + x - 2 = 0$. Our goal is to find the roots of this equation. 2. The quadratic equation is in the form $ax^2 + bx + c = 0$ where $a = 3

1. Problem: Find the inverse function of $f(x) = x + 60$. Step 1: Replace $f(x)$ by $y$: $y = x + 60$.

1. The problem is to solve the quadratic equation $3x^2 + x - 2 = 0$ for $x$. 2. We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=3$, $b=1$, and $c=-2

1. Problem: Given $p + \frac{1}{p} = 5$, find the value of $p^3 + \frac{1}{p^3}$. 2. Start with the identity: $$\left(p + \frac{1}{p}\right)^3 = p^3 + \frac{1}{p^3} + 3\left(p + \f

1. Stating the problem: Solve the quadratic equation $3xx^2 + x - 2 = 0$. 2. Rewrite the equation clearly: Since $xx^2 = x^3$, the equation becomes

1. We are asked to solve the quadratic equation $6ax^2 - 19a - 55 = 0$ for $x$. 2. First, identify the coefficients:

1. The problem asks: What kind of number are all in the x-coordinates of a logarithmic function? 2. A logarithmic function generally has the form $y = \log_a(x)$ where $a > 0$ and