🧮 algebra

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

1. Let's understand what an exponential function is. 2. An exponential function has the form $$f(x) = a^x$$ where $$a$$ is a positive constant not equal to 1.

1. The problem asks what kind of numbers the y-coordinates of an exponential function take. 2. An exponential function is generally defined as $$y = a^x$$ where $$a > 0$$ and $$a \

1. The problem asks for the equivalent value of the expression involving numbers 16, 64, 8, and 12. 2. We need to clarify the operation or function connecting these numbers but sin

1. The problem asks about the type of numbers that are all in the x-coordinates of an exponential function. 2. An exponential function is generally written as $$y = a^x$$ where $$a