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Homework 11-13

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4. Find a quadratic equation in standard form whose roots are 2 3 and 2 3 .i i− + − − 5. One root of the equation 23 8 0 is 4.x kx+ − = Find the other root and the value of k. 6. Show that 3 7i+ is a root of 2 6 58 0x x− + = Other helpful theorems:

If a quadratic equation, 2 0ax bx c+ + = , 0,a ≠ has real coefficients, then any complex zeros occur in conjugate pairs. That is, if a + bi is a zero then so is a – bi and vice-versa.

If a quadratic equation, 2 0ax bx c+ + = , 0,a ≠ has rational coefficients, then any irrational zeros occur in conjugate pairs. That is, if a b c+ is a zero then so is a b c− and vice-versa.

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Homework:

1. Find a quadratic equation in standard form with integral coefficients, one of whose

roots is 7 3 54i+ .

2. Show that 2 6i− − is a root of 2 4 40 0.x x+ + =

3. Show that 5 2 5i− is a root of 2 10 45 0.x x− + =

4. State the sum and the product of the roots of the given equation.

(a) 23 6 1 0x x− − + = (b) 2 3 0x + =

(c) 2 5x x= (d) 22 5 3x x− = −

5. Find a quadratic equation in standard form with integral coefficients and having the

given numbers as its roots.

(a) -2,-9 (b) 3 1,2 5

(c) 12 (double root) (d) 5,-7

(e) 3 (f) 8 4 3− −

6. One root of the equation 2 23 0 is .3

x x k+ + = Find the other root and the value of k.

7. One root of the equation 2 32 8 0 is .2 2kx x− + = Find the other root and the value of k.

8. One root of the equation 2 48 0 is 4.x kx+ − = Find the other root and the value of k. 9. One root of the equation 22 5 0 is -1.x kx+ − = Find the other root and the value of k.