Question

Using the quadratic formula, determine how many roots the following quadratic has:

x² – 5 = √3x
a) One real root
b) Two real roots
c) Two complex roots
d) No real roots

Answer 1

Using the quadratic formula, the equation x² - 5 = √3x has two real roots after rearranging it to standard form x² - 13x + 25 = 0 and finding a positive discriminant. Option b is the correct answer.

Step-by-step explanation:

To determine how many roots the quadratic equation x² – 5 = √3x has using the quadratic formula, we first need to re-arrange the equation into the standard quadratic form ax² + bx + c = 0.

We can do this by squaring both sides of the equation, which eliminates the square root on the right side, and then rearrange the terms:

(x² - 5)² = (√3x)²
x² - 10x + 25 = 3x
x² - 13x + 25 = 0

Now that we have the standard form, we can use the quadratic formula, which is:

x = (-b ± √(b² - 4ac)) / (2a)

For this quadratic equation, a = 1, b = -13, and c = 25.

The discriminant is (b² - 4ac) = (-13)² - 4(1)(25) = 169 - 100 = 69.

Since the discriminant is positive, this tells us that we have two real roots.