Question

describe and correct the error in solving using the quadratic formula.

\\x^2 + 10x + 74 = 0\\
\\x = \frac{-10 \pm \sqrt{10^2 - 4(1)(74)}}{2(1)}\\
\\= \frac{-10 \pm \sqrt{-196}}{2}\\
\\= \frac{-10 \pm 14}{2}\\
\\= -12 \text{ or } 2\\

Explanation:

Identify the error in the work

The student incorrectly simplified the square root of a negative number:

$$\sqrt{-196} eq 14$$

The square root of a negative number yields an imaginary number:

$$\sqrt{-196} = \sqrt{-1 \cdot 196} = 14i$$

Apply the quadratic formula correctly

For the equation \(x^2 + 10x + 74 = 0\), the coefficients are \(a = 1\), \(b = 10\), and \(c = 74\):

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(74)}}{2(1)}$$

$$x = \frac{-10 \pm \sqrt{100 - 296}}{2}$$

$$x = \frac{-10 \pm \sqrt{-196}}{2}$$

Simplify the complex expression

Using the definition of the imaginary unit, simplify the radical:

$$x = \frac{-10 \pm 14i}{2}$$

Divide each term in the numerator by the denominator:

$$x = -5 \pm 7i$$

Answer:

Error Description:
The error occurred when simplifying \(\sqrt{-196}\). The student simplified it as \(14\) instead of \(14i\), neglecting the negative sign under the radical which yields complex solutions.

Correct Solution:

$$x = -5 \pm 7i$$