1) The value of c is 4
2) The value of q is 91
Step-by-step explanation:
In the quadratic equation ax² + bx + c = 0,
1)
∵ The equation is 5x² - 12x + c = 0
∴ a = 5 , b = -12 , c = c
- Use the first rule to find the sum of the roots
∴ The sum of the two roots = [tex]-\frac{(-12)}{5}=\frac{12}{5}[/tex]
∵ One of the roots is 5 times the other root
- Assume that the other root is x
∵ the other root is x
∴ The one of the roots = 5 × x = 5x
∴ The sum of the roots = x + 5x = 6x
- Equate the two expressions of the sum of roots
∴ 6x = [tex]\frac{12}{5}[/tex]
- Divide both sides by 6
∴ x = [tex]\frac{2}{5}[/tex]
∴ 5x = 5( [tex]\frac{2}{5}[/tex] ) = 2
∴ The two roots are [tex]\frac{2}{5}[/tex] and 2
Multiply the two roots to find their product
∵ The product of the two roots = [tex]\frac{2}{5}[/tex] × 2 = [tex]\frac{4}{5}[/tex]
- By using the second rule above
∵ The product of the two roots = [tex]\frac{c}{5}[/tex]
- Equate the two expressions of the product of the roots
∴ [tex]\frac{c}{5}[/tex] = [tex]\frac{4}{5}[/tex]
- Multiply both sides by 5
∴ c = 4
The value of c is 4
2)
Assume that the two roots of the equation are m and n
∵ The equation is x² - 20x + q = 0
∴ a = 1 , b = -20 , c = q
∴ The sum of the roots = [tex]-\frac{-20}{1}=20[/tex]
- By using the first rule above
∵ m + n = 20 ⇒ (1)
∵ The difference between the roots is 6
∴ m - n = 6 ⇒ (2)
We have a system of equations let us solve it
Add equations (1) and (2) to eliminate n
∴ 2m = 26
- Divide both sides by 2
∴ m = 13
- Substitute the value of m in equation (1) to find n
∵ 13 + n = 20
- Subtract 13 from both sides
∴ n = 7
∴ The roots of the equations are 7 and 13
Find the product of the two roots
∵ The product of the two roots = 13 × 7 = 91
- By using the second rule above
∵ The product of the two roots = [tex]\frac{q}{1}[/tex] = q
- Equate the two expressions of the product of the roots
∴ q = 91
The value of q is 91
Learn more:
You can learn more about the system of equations in brainly.com/question/2115716
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