Find the discriminant and identify the best description of the equation's root(s). x2+5=sqrt(7x-2)

Question

Find the discriminant and identify the best description of the equation's root(s).

 x2+5=sqrt(7x−2)

 

a. 1 real solution

b. 2 complex solutions

c. 2 real solutions

d. 1 complex solution

e. 1 real and 1 complex solution 

 

Answer 1

First get rid of the square-root by raising both sides of equation to the 2nd power.

 

(x² + 5)² = 7x - 2

 

x⁴ + 10x² = 7x - 2

 

 

Next, make one side equal to zero.

 

x⁴ + 10x² - 7x + 2 = 0

 

We have a polynomial whose degree is 4 because the first term is x⁴.  The term after that is x².  These two terms graph the same way, so we will take x² as the first term, making the polynomial of degree 2.

 

If the formula used to find the roots of the quadratic equation is (-b ± √(b² - 4ac)) / 2a,

 

where:

a = 10

b = -7

c = 2

 

then the discriminant is the square-root part in the formula:  √(b² - 4ac).

 

If b² < 4ac, then the value under the square-root is negative and will be imaginary.

If b² ≥ 4ac, then the value under the square-root is positive and will be real.

 

 

Let's find b² and 4ac using the parameters above.

 

b² = (-7)² = 49

4ac = 4(10)(2) = 80

 

49 < 80.  Therefore, you will have two complex solutions since the formula has ±.

 

Your answer is (b).