1.1 Los op vir $x$: 1.1.1 $x (7 + x) = 0$ 1.1.2 $4x^2 - 5x - 4 = 0$ (korrek tot TWEE desimale plekke) 1.1.3 $2x^2 - 8 > 0$ 1.2 Los op vir $x$ en $y$ indien: $y = 5x - 2$ en $y = x^3 + 4x - 8$ 1.3 Die diagram hieronder toon die beweging van 'n suier binne die motor-ein se silinder - NSC Technical Mathematics - Question 1 - 2022 - Paper 1

Using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 4$, $b = -5$, and $c = -4$:

$x = \frac{5 \pm \sqrt{(-5)^2 - 4(4)(-4)}}{2(4)} = \frac{5 \pm \sqrt{25 + 64}}{8} = \frac{5 \pm \sqrt{89}}{8}$

Calculating the roots:

• $x_1 \approx 1.80$
• $x_2 \approx -0.55$

Thus, the solutions are $x \approx 1.80$ and $x \approx -0.55$.

To find the critical points, we set $2x^2 - 8 = 0$:

$2(x^2 - 4) = 0 \implies (x - 2)(x + 2) = 0$

Thus, $x = 2$ or $x = -2$. We analyze the intervals to determine the solution set:

• For $x < -2$, the expression is positive.
• For $-2 < x < 2$, the expression is negative.
• For $x > 2$, the expression is positive.

Thus, the solution set is $(-\infty, -2) \cup (2, \infty)$.

To find $x$ and $y$, we set the equations equal to each other:

$5x - 2 = x^3 + 4x - 8$

Rearranging leads to:

$x^3 - x + 6 = 0$

By testing potential rational roots, we find:

• At $x = -2$: $(-2)^3 - (-2) + 6 = 0$

Thus, $x = -2$. We can use synthetic division to factor the cubic and find other roots as $x = 3$ and $x = -2$:

So $y = 5(-2) - 2 = -12$ and $y = 5(3) - 2 = 13$. Therefore, the solutions are:

• $x = -2$ leads to $y = -12$
• $x = 3$ leads to $y = 13$.

Starting with the formula for the slagvolume:

$SV = \frac{\text{π} d^2 \times L}{4}$

To isolate $L$, we rearrange the equation:

$L = \frac{4SV}{\text{π}d^2}$

Using the formula derived above:

Substituting $SV = 1 020,5$ and $d = 10$:

$L = \frac{4(1 020,5)}{\text{π}(10)^2} \approx \frac{4 082}{100\text{π}} \approx 13 \, \text{cm}$

The values given are:

• $P = 1 010$
• $Q = 10 000$

These could represent various properties like pressure and volume, expressed in appropriate units depending on the context.