I can help break down the math problems in the image you provided. Let me go through each question and solve them one by one.

1. Sebutkan Ciri-ciri Persamaan Kuadrat:

Characteristics of a quadratic equation include:

• The equation has the general form: $ax^2 + bx + c = 0$, where $a \neq 0$.
• The highest degree of the variable (typically $x$) is 2.
• It may have 0, 1, or 2 real solutions depending on the discriminant $\Delta = b^2 - 4ac$.

2. Tentukan apakah persamaan berikut termasuk persamaan kuadrat atau bukan!

• (a) $x^2 = 0$: Yes, this is a quadratic equation because it has the form $x^2 + 0x + 0 = 0$.
• (b) $2x + 5 - x^2 = 0$: Yes, this is a quadratic equation as it can be written as $-x^2 + 2x + 5 = 0$, which fits the form $ax^2 + bx + c = 0$.
• (c) $3x^3 = 0$: No, this is not a quadratic equation since the highest power is 3.
• (d) $4x^2 + 5x = 0$: Yes, this is a quadratic equation because it can be written as $4x^2 + 5x + 0 = 0$.
• (e) $x + 5x - 6 = 0$: No, this is not a quadratic equation because it can be simplified to $6x - 6 = 0$, which is linear.

3. Tentukan nilai a, b, dan c dari persamaan kuadrat berikut!

For each quadratic equation, identify the coefficients $a$, $b$, and $c$:

• (a) $x^2 + x = 0$: Here, $a = 1$, $b = 1$, and $c = 0$.
• (b) $2x^2 + 3x - 5 = 0$: Here, $a = 2$, $b = 3$, and $c = -5$.
• (c) $-3x^2 + 2x - 3x = 0$: Simplify this to $-3x^2 - x = 0$, so $a = -3$, $b = -1$, and $c = 0$.

4. Tentukan faktor dari persamaan kuadrat:

• (a) $x^2 = 0$: Factor as $x(x) = 0$, so the factors are $x(x)$.
• (b) $x^2 - 10 = 0$: Factor as $(x - \sqrt{10})(x + \sqrt{10}) = 0$.
• (c) $x^2 + 6x + 8 = 0$: Factor as $(x + 2)(x + 4) = 0$.
• (d) $x^2 - x + 1 = 0$: This cannot be factored using real numbers because the discriminant $\Delta = b^2 - 4ac$ is negative.
• (e) $x^2 + 11x + 16 = 0$: Factor as $(x + 8)(x + 2) = 0$.

5. Tentukan akar-akar dari:

• (a) $x^2 = 0$: The root is $x = 0$.
• (b) $x^2 - x = 0$: Factor as $x(x - 1) = 0$, so the roots are $x = 0$ and $x = 1$.
• (c) $2x^2 + 11x + 1 = 0$: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots. Here, $a = 2$, $b = 11$, and $c = 1$. The discriminant $\Delta = 11^2 - 4(2)(1) = 121 - 8 = 113$, so the roots are: $x = \frac{-11 \pm \sqrt{113}}{4}$

6. Tentukan nilai dari $x_1, x_2$, dan $x_3$ dari:

• (a) $x^3 + 6x + 8 = 0$: This is a cubic equation, so you will need to use methods like synthetic division or trial and error to find one root, then factor the quadratic part.
• (b) $5x^2 + 10x + 1 = 0$: Apply the quadratic formula for roots.
• (c) $x^2 + 4x + 4 = 0$: Factor as $(x + 2)^2 = 0$, so the root is $x = -2$ (a double root).

Would you like any specific step explained in more detail, or any of these topics expanded?

Here are five follow-up questions to deepen understanding:

1. What are the properties of the discriminant, and how do they determine the nature of the roots?
2. How do we factor a quadratic equation if the coefficients are large numbers or fractions?
3. Can we always solve cubic equations using factoring? What are alternative methods?
4. How is the quadratic formula derived from the general quadratic equation?
5. What is the significance of double roots in quadratic equations?

Tip: Always check the discriminant first when solving quadratic equations—it tells you whether the roots are real or complex.