Answered

Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Question #9: Given the quadratic equation below, solve for the roots. Round your answer to two decimal places.

[tex]\[ x^2 + 4x + 1 = 0 \][/tex]

A. -0.19 and 3.71
B. 0.35 and -3.11
C. -0.27 and -3.73
D. -0.26 and -3.5

Sagot :

To solve the quadratic equation [tex]\( x^2 + 4x + 1 = 0 \)[/tex], we will use the quadratic formula, which is:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 1 \)[/tex]

First, let's calculate the discriminant ([tex]\( \Delta \)[/tex]), which is given by:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:

[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12 \][/tex]

The discriminant is 12.

Next, we use the quadratic formula to find the roots. We need to calculate each root separately using the positive and negative forms of the [tex]\(\pm\)[/tex] sign in the formula.

The first root ([tex]\( x_1 \)[/tex]) is calculated as:

[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-4 + \sqrt{12}}{2 \cdot 1} = \frac{-4 + \sqrt{12}}{2} \][/tex]

Solving this, we get:

[tex]\[ x_1 = \frac{-4 + 3.4641016151377544}{2} = \frac{-0.5358983848622456}{2} = -0.2679491924311228 \][/tex]

Rounding [tex]\( x_1 \)[/tex] to two decimal places:

[tex]\[ x_1 \approx -0.27 \][/tex]

The second root ([tex]\( x_2 \)[/tex]) is calculated as:

[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-4 - \sqrt{12}}{2 \cdot 1} = \frac{-4 - \sqrt{12}}{2} \][/tex]

Solving this, we get:

[tex]\[ x_2 = \frac{-4 - 3.4641016151377544}{2} = \frac{-7.4641016151377544}{2} = -3.732050807568877 \][/tex]

Rounding [tex]\( x_2 \)[/tex] to two decimal places:

[tex]\[ x_2 \approx -3.73 \][/tex]

Therefore, the roots of the quadratic equation [tex]\( x^2 + 4x + 1 = 0 \)[/tex], rounded to two decimal places, are:

[tex]\[ \boxed{-0.27 \text{ and } -3.73} \][/tex]