To find the sum and product of the roots of each given quadratic equation, we use the following formulas:

• Sum of the roots: $-\frac{b}{a}$
• Product of the roots: $\frac{c}{a}$

where the quadratic equation is in the form $ax^2 + bx + c = 0$.

Let's go through each equation.

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1. $x^2 + 2x + 1 = 0$

• Here, $a = 1$, $b = 2$, and $c = 1$.
• Sum of the roots: $-\frac{b}{a} = -\frac{2}{1} = -2$
• Product of the roots: $\frac{c}{a} = \frac{1}{1} = 1$

Answer:

• Sum of the roots: $-2$
• Product of the roots: $1$

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2. $x^7 + 10x + 16 = 0$ (Not a quadratic equation)

This is not a quadratic equation because the highest power of $x$ is 7. Therefore, we cannot use the formulas for the sum and product of roots designed for quadratic equations.

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3. $x^2 + 10x + 16 = 0$

• Here, $a = 1$, $b = 10$, and $c = 16$.
• Sum of the roots: $-\frac{b}{a} = -\frac{10}{1} = -10$
• Product of the roots: $\frac{c}{a} = \frac{16}{1} = 16$

Answer:

• Sum of the roots: $-10$
• Product of the roots: $16$

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4. $4x^2 + 8x - 16 = 0$

• Here, $a = 4$, $b = 8$, and $c = -16$.
• Sum of the roots: $-\frac{b}{a} = -\frac{8}{4} = -2$
• Product of the roots: $\frac{c}{a} = \frac{-16}{4} = -4$

Answer:

• Sum of the roots: $-2$
• Product of the roots: $-4$

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5. $8x^2 - 4x - 24 = 0$

• Here, $a = 8$, $b = -4$, and $c = -24$.
• Sum of the roots: $-\frac{b}{a} = -\frac{-4}{8} = \frac{1}{2}$
• Product of the roots: $\frac{c}{a} = \frac{-24}{8} = -3$

Answer:

• Sum of the roots: $\frac{1}{2}$
• Product of the roots: $-3$

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Let me know if you need more details on these calculations or if you have any questions.

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Relative Questions

1. How do we derive the formula for the sum and product of the roots?
2. Why can't we apply the same method to polynomials of degree higher than 2?
3. How would we factor each quadratic equation directly to verify the roots?
4. What is the significance of the discriminant in these quadratic equations?
5. Can the sum and product of roots help in sketching the graph of the quadratic equation?

Tip: To factor quadratic expressions, look for numbers that multiply to the constant term $c$ and add to the middle coefficient $b$.