Question

Marcus used the quadratic formula, x=−b±b2−4ac√2a,

to solve the equation 0=x2+5x+17
as follows.

Step 1: x=−5±52−4(1)(17)√2(5)
Step 2: x=−5±25−68√10
Step 3: x=−5±−43√10
Step 4: x=−12±43√10i
Did Marcus make a mistake? If so, where?

Answer 1

In summary, the mistake made by Marcus is in Step 2, where he miscalculated the value of the discriminant. The correct value of the discriminant is -43!

Explanation:

Yes, Marcus made a mistake in Step 2 of his solution!

In Step 2, Marcus wrote:

x = -5 ± 25 - 68√10.

The mistake is in the expression 25 - 68√10. According to the quadratic formula, the discriminant is represented as b^2 - 4ac under the square root symbol. In this case, a = 1, b = 5, and c = 17.

Let's calculate the correct value of the discriminant:

b^2 - 4ac = (5)^2 - 4(1)(17) = 25 - 68 = -43.

So, the correct value of the discriminant is -43.

Now, let's rewrite Step 2 correctly:

x = -5 ± √(-43)√10.

Since the discriminant is negative (-43), we have to use the imaginary unit "i" to represent the square root of -43.

Therefore, the correct solution is:

x = -5 ± √(-43)√10i.

In Step 4, Marcus correctly wrote:

x = -12 ± 43√10i

Answer 2

This is what his steps should look like.

`\text{x} = (-b\pm√(b^2-4ac))/(2a)\\\\\text{x} = (-5\pm√((5)^2-4(1)(17)))/(2(1))\\\\\text{x} = (-5\pm√(25 - 68))/(2)\\\\\text{x} = (-5\pm√(-43))/(2)\\\\\text{x} = (-5\pm i√(43))/(2)\\\\`