+46 1. Fill in the blanks to make a quadratic whose roots are 5 and -3.
\(x^2+\boxed{\color{white}ABC}x+\boxed{\color{white}ABC} \)
2. Let \(a\) and \(b\) be the roots of the quadratic Find the quadratic whose roots are \(a^2\) and \(b^2\).
\(x^2+\boxed{\color{white}ABC}x+\boxed{\color{white}ABC} \)
3. Let \(a\) and \(b\) be the roots of the quadratic \(2x^2 - 8x + 7 = 0\). Compute \(a^3 + b^3.\)
4. Fill in the blanks to make the equation true.
\((\boxed{\color{white}ABC}x+\boxed{\color{white}ABC})(\boxed{\color{white}ABC}x+\boxed{\color{white}ABC})=\boxed{\color{white}ABC}x^2+\boxed{\color{white}ABC}x+\boxed{\color{white}ABC}\)
Options:
+128707 1. (x - 5) (x + 3) = 0
x^2 - 2 x + - 15 = 0
3. 2x^2 -8x + 7 = 0
Product of roots = ab = 7/2
2ab = 7 (1)
Sum of roots = a + b = - (-8) / 2 = 4
Square both sides
a^2 + 2ab + b^2 = 16 (2)
Sub (1) into (2)
a^2 + 7 + b^2 =16
a^2 + b^2 = 9
a^3 + b^3 =
(a + b) ( a^2 - ab + b^2) =
(a + b) ( a^2 + b^2 - ab) =
(4) (9 - 7/2) =
(4) ( 11/2) =
22
