Los op vir $x$: 1.1.1 $x^2 - 7x + 12 = 0$ 1.1.2 $x(3x + 5) = 1$ (korrekt tot TWEE desimale syfers) 1.1.3 $x^2 - 2x + 15$ 1.1.4 $ ext{ } \\sqrt{2(1-x)} = x - 1 $ 1.2 Los gelikdtyd vir $x$ en $y$ op: $3^{x+y} = 27$ $ x^2 + y^2 = 17$ 1.3 Bepaal, sonder die gebruik van 'n sakrekenaar, die waarde van: $rac{1}{ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \\sqrt{1 + \rac{1}{2}} + \rac{1}{ ext{ } \sqrt{2 + \sqrt{3}} + \rac{1}{ ext{ } \sqrt{3 + \sqrt{4}} + .. - NSC Mathematics - Question 1 - 2023 - Paper 1

To solve for $x$:

1. Rearrange to standard form: $3x^2 + 5x - 1 = 0$.
2. Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 3$, $b = 5$, and $c = -1$.
3. Calculate the discriminant: $b^2 - 4ac = 5^2 - 4(3)(-1) = 25 + 12 = 37$.
4. Substitute into the formula: $x = \frac{-5 \pm \sqrt{37}}{2(3)} = \frac{-5 \pm \sqrt{37}}{6}$.
5. The approximate solutions are $x \approx -0.18$ and $x \approx -1.85$.

To analyze this quadratic:

1. Rewrite in standard form: $x^2 - 2x + 15$.
2. Calculate vertices or check for real roots by using the discriminant: $b^2 - 4ac = 2^2 - 4(1)(15) = 4 - 60 = -56$. Since the discriminant is negative, there are no real roots.

To solve the equation:

1. Square both sides: $2(1-x) = (x - 1)^2$.
2. Expand and simplify: $2 - 2x = x^2 - 2x + 1$.
3. Rearrange to standard form: $x^2 - 1 = 0$.
4. Factor: $(x + 1)(x - 1) = 0$.
5. Solutions are $x = 1$ and $x = -1$.

1. From the first equation, write $27$ as $3^3$: $3^{x+y} = 3^3$, hence $x + y = 3$.
2. Substitute $y = 3 - x$ into the second equation: $x^2 + (3-x)^2 = 17$.
3. Expanding gives $x^2 + 9 - 6x + x^2 = 17$ leading to $2x^2 - 6x - 8 = 0$.
4. Dividing everything by 2 gives: $x^2 - 3x - 4 = 0$. Solving this quadratic: $(x - 4)(x + 1) = 0$.
5. Therefore, $x = 4$ or $x = -1$, leading to $y= -1$ or $y = 4$.

To evaluate:

1. The general term is written as $\frac{1}{\sqrt{n}}$.
2. Recognizing the series structure: Rationalize each term and simplify: $\sqrt{1 + \frac{1}{2}}$ progressively till $\sqrt{99} + \frac{1}{\sqrt{100}}$.
3. Through calculation, combine the results giving $= -1 + \sqrt{2} - \sqrt{3} + \cdots = 9$.