Answer:
1) x² - 6x + 58 = 0
2) x² - 12x + 45 = 0
3) x² - 10x + 23 = 0
4) x² - 2x - 48 = 0
Step-by-step explanation:
1-4 The form the quadratic equation with its roots:
ax² + -b/a x + c/a = 0 ⇒ b is the sum of the roots and c is the product of the roots.
If the roots are irrational numbers or complex number then they are conjugate
Ex: 1] irrational [tex](2+\sqrt{3})(2-\sqrt{3})[/tex] ⇒ their sum and product are
rational numbers
2] complex (5 - 3i)(5 + 3i) ⇒ their sum and product are rational numbers
1) ∵ The first root is 3 + 7i ⇒ ∴ the second root is 3 - 7i
∵ their sum = 3 + 7i + 3 - 7i = 6
∵ their product = (3 + 7i)(3 - 7i) = 9 - 49i² = 9 + 49 = 58⇒(i² = -1)
∴ The equation⇒ x² - 6x + 58 = 0
2) ∵ The first root is 6 - 3i ⇒ ∴ the second root is 6 + 3i
∵ their sum = 6 - 3i + 6 + 3i = 12
∵ their product = (6 - 3i)(6 + 3i) = 36 - 9i² = 36 + 9 = 45⇒(i² = -1)
∴ The equation⇒ x² - 12x + 45 = 0
3) ∵ The first root is 5 + √2 ⇒ ∴ the second root is 5 - √2
∵ their sum = 5 + √2 + 5 - √2 = 10
∵ their product = (5 + √2)(5 - √2) = 25 - 2 = 23
∴ The equation⇒ x² - 10x + 23 = 0
4) ∵ The first root is -6 and the second root is 8
∵ their sum = -6 + 8 = 2
∵ their product = (-6)(8) = -48
∴ The equation⇒ x² - 2x - 48 = 0
5-8 Solving equations:
1) x² - 6x + 58 = 0
a = 1 , b = -6 , c = 58
x = [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}=\frac{--6+\sqrt{36-(4)(1)(58)} }{(2)(1)}\\ =\frac{6+\sqrt{196} }{2} =\frac{6+14i}{2}=3+7i[/tex]
2) x² - 12x + 45 = 0
a = 1 , b = -12 , c = 45
x = [tex]\frac{--12+\sqrt{144-(4)(1)(45)} }{(2)(1)}=\frac{12+\sqrt{-36} }{2}\\=\frac{12+6i}{2}=6+3i[/tex]
3) x² - 10x + 23 = 0
a = 1 , b = -10 , c = 23
x = [tex]\frac{--10+\sqrt{100-(4)(1)(23)} }{(2)(1)}=\frac{10+\sqrt{8} }{2}\\ =\frac{10+2\sqrt{2} }{2}=5+\sqrt{2}[/tex]
4) x² - 2x - 48 = 0 ⇒ using factorization
(x - 8)(x + 6) = 0
x - 8 = 0⇒ x = 8
x + 6 = 0⇒ x = -6
