# Tutor profile: Nancy L.

## Questions

### Subject: Pre-Algebra

If the length of of a rectangle is 10 and the width is 5, what is the area and the perimeter?

The formula the area of a rectangle is width times length and the perimeter is 2 times the length plus the width. Area: $$length * width $$ Perimeter: $$2 *(length + width)$$ Therefore in this question the area would be 10 * 5 = 50 and the perimeter would be 2 * (10 + 5) = 30

### Subject: Calculus

Find the derivative of $$f(x) = 3x^2 + 2x + 1$$

The power rule says $$f(x) = x^n$$ is $$f'(x) = nx^{n-1}$$ We also have some nice addition and constant multiplication properties: $$(f(x) + g(x))' = f'(x) + g'(x)$$ This equation shows that the derivative of two function added together is equal to the addition of derivatives of each function. $$(cf(x))' = cf'(x)$$ This equation shows that the derivative of a function that is multiplied by a constant is equal to the derivative of that function times the constant. We can use these formula to solve the above question. We have $$(3x^2)' = 3 * (x^2)' = 3* 2x = 6x$$ $$(2x)' = 2(x)' = 2$$ and the derivative of a constant is just $$0$$ so $$1' = 0$$ Then we can add the three parts together and get $$f(x)' = 6x + 2$$

### Subject: Algebra

Solve $$2x^2 + 4x + 1 = 0 $$

There are two ways of solving this problem 1. Formulaic 2. Perfect Square 1. The formula for solving a typical equation $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ . In this question, a = 2, b = 4 and c = 1. Therefore plugging the three values into the formula would give $$x = \frac{-4 \pm \sqrt{4^2-4*2*1}}{2*2} = \frac{-4 \pm \sqrt{16-8}}{4} = \frac{-4 \pm 2\sqrt{2}}{4} = \frac{-2 \pm \sqrt{2}}{2} = -1 \pm\frac{\sqrt{2}}{2} $$ 2. The perfect square formula tells us that $$x^2 + 2x + 1 = (x+1)^2$$. Therefore we can manipulate the original equation to the equation above. We can divide both sides by 2 and get $$ x^2 + 2x + \frac{1}{2} = 0$$. Now we can add \frac{1}{2} to both sides of the equation and get $$ x^2 + 2x + 1 = \frac{1}{2}$$. Now, plugging in the perfect square formula, we get $$(x+1)^2 = \frac{1}{2}$$. Taking square roots of both sides of the equation will give us $$x + 1 = \pm\sqrt{\frac{1}{2}}$$. We can now subtract 1 from both sides of the equation and get $$x = \pm\sqrt{\frac{1}{2}} -1$$ which simplified is equal to $$x = \pm\frac{\sqrt{2}}{2} -1$$

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