describe the nature of the roots of the equation 7x^(2)=4x+1

Question

a. one real root

b. two real roots

c. two imaginary roots

d. cannot be determined

Answer 1

Put the equation in standard form by bringing the 4x + 1 to the left side.

7x² - 4x - 1 = 0

We use the discriminant to determine the nature of the roots of a quadratic equation. The discriminant is the expression underneath the radical in the quadratic formula: b² - 4ac. 

b² - 4ac          In this case, a = 7, b = -4, and c = -1

(-4)² - 4(7)(-1)

16 + 28 = 44

Now here are the rules for determining the nature of the roots:

(1) If the discriminant = 0, then there is one real root (this omits the ± from the quadratic formula, leaving only one possible solution)

(2) If the discriminant > 0, then there are two real roots (this keeps the ±, giving you two solutions)

(3) If the discriminant < 0, then there are two imaginary roots (this means there is a negative under the radical, making the solutions imaginary)

44 > 0, so there are two real roots (choice B)