Los op vir $x$: 1.1.1 $$3x^2 + 5x = 0$$ 1.1.2 $$4x^2 + 3x - 5 = 0$$ (answere korrek tot TWEE desimale plekke) 1.1.3 $$(x-1)^2-9 eq 0$$ 1.1.4 $$5^2 - 5^x = 0$$ 1.1.5 $$\frac{x}{\sqrt{20-x}} = 1$$ Los gelyktydig vir $x$ en $y$ op: $$x+y=9$$ en$$2x^2-y^2=7$$ Gegee: $$P = (1-a)(1+a)(1+a^2)(1+a^4)(1+a^{12})$$ en$$T = (1+a)(1+a^2)(1+a^4)$$ Bepaal die waarde van $$P \times T$$ in terme van $a$. - NSC Mathematics - Question 1 - 2024 - Paper 1

We will use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 4$, $b = 3$, and $c = -5$.

Calculating the discriminant: $D = 3^2 - 4(4)(-5) = 9 + 80 = 89.$

Thus, $x = \frac{-3 \pm \sqrt{89}}{8}.$

Calculating the two roots gives us approximate values of $x \approx 0.80$ and $x \approx -1.55$. Both should be reported to two decimal places.

Rearranging gives: $(x-1)^2 \neq 9.$ Taking the square root of both sides leads to: $x - 1 \neq \pm 3.$ Therefore, we have:

1. $x \neq 4$
2. $x \neq -2.$

Rearranging gives: $5^2 = 5^x.$ This implies that: $x = 2.$

Additionally, when $5^x = 0$ there are no solutions since the function does not equal zero.

Isolate the surd: $x = \sqrt{20-x}.$ Squaring both sides results in: $x^2 = 20 - x.$ Rearranging gives: $x^2 + x - 20 = 0.$ Factoring this we find: $(x+5)(x-4) = 0.$ So, $x = 4$ or $x = -5.$

From the first equation, we can express $y$ in terms of $x$: $y = 9 - x.$

Substituting into the second equation: $2x^2 - (9-x)^2 = 7.$ This simplifies to: $2x^2 - (81 - 18x + x^2) = 7.$ Combining like terms leads to: $x^2 + 18x - 88 = 0.$ Applying the quadratic formula: $x = \frac{-18 \pm \sqrt{(18)^2 - 4(1)(-88)}}{2(1)}.$ Calculating the solutions gives us two $x$-values. We then substitute back into $y = 9 - x$ to find the corresponding y-values.

To find $P \times T$, we first express both in a product: $P \times T = [(1-a)(1+a)(1+a^2)(1+a^4)(1+a^{12})] \times [(1+a)(1+a^2)(1+a^4)].$

This expands and factors appropriately, leading to the final expression simplified in terms of $a$.