Exercises 18-20: In each of these problems, an equation and one of its roots are given. Find:
a the value of k
b the second root
18.5 is a root of x^2- 7x + k = 0
19. 5 is the root of x^2-3x+k=0
20.7 is the root of x^2-3x= -k
Please help! It’s three questions

(18) a. The value of k is 10.

b. The second root is 2.

(19) a. The value of k is -10.

b. The second root is -2.

(20) a. The value of k is -28.

b. The second root is -4.

Step-by-step explanation

18. One of the roots is 5, :. x = 5

Substitute x = 5 into the quadratic equation to obtain the value of k.

x² - 7x + k = 0

5² - 7(5) + k = 0

25 - 35 + k = 0

-10 + k = 0

k = 0 + 10 = 10

Factorise the quadratic equation to obtain the value of the second root.

x² - 7x + 10 = 0

x² - 5x - 2x + 10 = 0

(x² - 5x) - (2x + 10) = 0

x(x - 5) -2(x - 5) = 0

(x - 5)(x - 2) = 0

x = 5 or 2

The second root is 2.

19. One of the roots is 5, :. x = 5

Substitute x = 5 into the quadratic equation to obtain the value of k.

x² - 3x + k = 0

5² - 3(5) + k = 0

25 - 15 + k = 0

10 + k = 0

k = 0 - 10 = -10

Factorise the quadratic equation to obtain the value of the second root.

x² - 3x + (-10) = 0

x² - 3x - 10 = 0

x² - 5x + 2x - 10 = 0

(x² - 5x) + (2x - 10) = 0

x(x - 5) +2(x - 5) = 0

(x - 5)(x + 2) = 0

x = 5 or -2

The second root is -2.

20. One of the roots is 7, :. x = 7

Substitute the value of x into the quadratic equation to obtain the value of k.

x² - 3x = -k

7² - 3(7) = -k

49 - 21 = -k

28 = -k

k = -28

Factorise the quadratic equation to obtain the value of the second root.

x² - 3x = -k

x² - 3x = 28

x² - 3x - 28 = 0

x² - 7x + 4x - 28 = 0

(x² - 7x) + (4x - 28) = 0

x(x - 7) +4(x - 7) = 0

(x - 7)(x + 4) = 0

x = 7 or -4

The second root is -4.

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