Solve for $x$: 1.1.1 $-2x(x + a)(3 - x) = 0$ 1.1.2 $2x = 6 - x^2$ (correct to TWO decimal places) 1.1.3 $5x(x - 3) \, <= 0$ and then represent the solution on a number line - NSC Technical Mathematics - Question 1 - 2018 - Paper 1

Rearranging the equation gives us:

$x^2 + 2x - 6 = 0$

Using the quadratic formula, where $a = 1$, $b = 2$, and $c = -6$:

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

Substituting the values, we find:

$x = \frac{-2 \pm \sqrt{4 + 24}}{2} = \frac{-2 \pm \sqrt{28}}{2} = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7}$

Calculating the approximations:

1. $x \approx 1.65$
2. $x \approx -3.65$ (Not valid in this context.)

Therefore, the accepted value is $x \approx 1.65$.

To analyze the inequality $5x(x - 3) \leq 0$, first find the critical values:

1. $x = 0$
2. $x = 3$

Next, we test intervals created by these critical values:

1. For $x < 0$: (choose $x = -1$) → $5(-1)(-1 - 3) = 20 > 0$
2. For $0 < x < 3$: (choose $x = 1$) → $5(1)(1 - 3) = -10 < 0$
3. For $x > 3$: (choose $x = 4$) → $5(4)(4 - 3) = 20 > 0$

Thus, the solution is $x \in [0, 3]$, which should be represented on a number line.

From $y - 2x = -7$, rearranging gives: $y = 2x - 7$

Substituting this into $x^2 + xy + y^2 = 21$:

$x^2 + x(2x - 7) + (2x - 7)^2 = 21$

Expanding and simplifying:

t$x^2 + 2x^2 - 7x + 4x^2 - 28x + 49 = 21$ $7x^2 - 35x + 28 = 0$

Dividing through by 7: $x^2 - 5x + 4 = 0$

Factoring gives: $(x - 4)(x - 1) = 0$

Thus, $x = 4$ or $x = 1$. Plugging these back into $y = 2x - 7$:

• For $x = 4$: $y = 2(4) - 7 = 1$
• For $x = 1$: $y = 2(1) - 7 = -5$

The dimensions of the squares are $(4, 1)$ and $(1, -5)$.

To isolate $L$, we first multiply both sides by 12: $12E = ML^2$

Now, divide both sides by $M$: $L^2 = \frac{12E}{M}$

Taking the square root: $L = \sqrt{\frac{12E}{M}}$

Using the formula derived: $L = \sqrt{\frac{12(8.3 \times 10^2)}{1.6 \times 10^3}}$

Calculating: $L = \sqrt{\frac{99.6 \times 10^2}{1.6 \times 10^3}} = \sqrt{\frac{99.6}{1.6} \times 10^{-1}} \approx \sqrt{62.25 \times 10^{-1}} \approx 0.79 \, m$

To convert 36 to binary, we can repeatedly divide by 2 and record the remainders:

• 36 ÷ 2 = 18, remainder 0
• 18 ÷ 2 = 9, remainder 0
• 9 ÷ 2 = 4, remainder 1
• 4 ÷ 2 = 2, remainder 0
• 2 ÷ 2 = 1, remainder 0
• 1 ÷ 2 = 0, remainder 1

Reading from bottom to top, we get 36 in binary as $100100_2$.