Misc

Question 1

A) What are “roots”?

B) What are the roots of 2x² + 3x = 2?

Answer 1

A) Roots translates to “x =”

Roots are the values for x that solve a quadratic.

If we have the equation (x - r) (x - s) = 0, the roots would be r and s.

B) Subtract 2 to get 0 on the right side: 2x² + 3x - 2 = 0

Factor: (2x - 1) (x + 2) = 0

Roots are the solutions for x: 1/2 and -2

Question 2

If r and s are the roots of the equation x^2 + bx + c = 0, where b and c are constants, is rs < 0 ?

(1) b < 0
(2) c < 0

Answer 2

“r and s are the roots” translates to (x - r) (x - s) = 0

Therefore, c = -r * -s = rs

1) Insufficient –> rs could be positive or negative.

Example: (x-1) (x-2) = 0 –> b = -3, rs=2

(x+1) (x-2) = 0 –> b = -1, rs = -2

2) Sufficient –> c = rs, so if c < 0, rs < 0

B - (2) Alone

Question 3

Which of the following equations has -1 + √2 as one of its roots?

A) x² + 2x – 1 = 0
B) x² – 2x + 1 = 0
C) x² + 2x + 1 = 0
D) x² – 2x – 1 = 0
E) x² – x – 1= 0

Answer 3

“roots” translates to x = -1 + √2

x + 1 = √2

square both sides: (x + 1)² = 2

x² + 2x + 1 = 2

A) x² + 2x - 1 = 0