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V.
1. [tex]ax^2 + bx + c = 0[/tex] is a quadratic equation.

a. Write the roots of the equation.

b. If [tex]a[/tex] and [tex]β[/tex] are the roots of the equation, write the relations among [tex]a[/tex], [tex]b[/tex], [tex]a[/tex], and [tex]c[/tex].

c. Find the quadratic equation whose roots are 7 and −7.

2. The hypotenuse of a right-angled triangle is 10 cm, and the other two sides are in the ratio of 3:4.

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Note: Translations and relations among variables are preserved; some typographical errors and unclear symbols were corrected.

Sagot :

GinnyAnswer
Certainly! Let's address the last question regarding the right-angled triangle with a hypotenuse of 10 cm and the other two sides in a ratio of 3:4.

### Given:
- Hypotenuse (c): 10 cm
- Ratio of the other two sides (a:b): 3:4

### Find the lengths of the other two sides of the triangle.

### Solution:

1. Ratios and Variables:
- Let the two sides be [tex]\(3x\)[/tex] and [tex]\(4x\)[/tex] where [tex]\(x\)[/tex] is a common factor.

2. Using the Pythagorean Theorem:
- According to the Pythagorean theorem for a right-angled triangle:
[tex]\[ (3x)^2 + (4x)^2 = 10^2 \][/tex]

3. Calculating:
- Expanding the equation:
[tex]\[ 9x^2 + 16x^2 = 100 \][/tex]
- Combining like terms:
[tex]\[ 25x^2 = 100 \][/tex]
- Solving for [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = 4 \][/tex]
- Taking the square root of both sides:
[tex]\[ x = 2 \][/tex]

4. Finding the Side Lengths:
- Using [tex]\(x = 2\)[/tex], calculate the lengths of the sides:
[tex]\[ \text{Side } a = 3x = 3 \times 2 = 6 \text{ cm} \][/tex]
[tex]\[ \text{Side } b = 4x = 4 \times 2 = 8 \text{ cm} \][/tex]

### Conclusion:
The lengths of the other two sides of the right-angled triangle are 6 cm and 8 cm, respectively.
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