Question

\(s^2+10 s+8=0\)

Short answer

So, the roots of the equation \(s^2+10s+8=0\) are -3.87 and -2.13

Step-by-step solution

Identify the coefficients

First, it is necessary to identify the coefficients a, b, and c. In this equation, \(a = 1\), \(b = 10\), and \(c = 8\).

Substitute values into the quadratic formula

Now put the values of a, b and c in the quadratic formula. Therefore: \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) becomes: \(s = \frac{-10 \pm \sqrt{(10)^2 - 4*(1)*(8)}}{2*1}\)

Calculate the discriminant

Calculate the value under the square root, which is known as the discriminant (D): \(D = b^2 - 4*a*c = 100 - 32 = 68 \)

Compute the roots

Use the discriminant to compute the roots \(s1\) and \(s2\): \(s1 = \frac{-(-10) + \sqrt{68}}{2} = -3.87, s2 = \frac{-(-10) - \sqrt{68} }{2} = -2.13 \)

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation, which is typically in the form \( ax^2 + bx + c = 0 \). The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It allows us to solve for the unknown variable \( x \) using the three coefficients (a, b, and c) found in the quadratic expression.
The formula is derived from the process of completing the square and is invaluable for providing exact solutions whether the quadratic can be factored easily or not.

Coefficients

Coefficients in a quadratic equation are the numerical components of the terms. In the quadratic equation \( ax^2 + bx + c = 0 \), these coefficients are represented by:

• \( a \): the coefficient of \( x^2 \)
• \( b \): the coefficient of \( x \)
• \( c \): the constant term, or the coefficient of \( x^0 \)

In the example \( s^2 + 10s + 8 = 0 \), \( a = 1 \), \( b = 10 \), and \( c = 8 \).
Identifying these coefficients correctly is essential since they directly influence the shape and position of the parabola represented by the quadratic equation, as well as the calculation of roots.

Discriminant

The discriminant is a specific part of the quadratic formula under the square root, represented as \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation.
Once calculated, the value of the discriminant will indicate:

• If it's positive: two distinct real roots
• If it's zero: one real double root
• If it's negative: two complex roots

In our equation, the discriminant \( D = 10^2 - 4 \times 1 \times 8 = 68 \), which is positive, confirming that there are two distinct real roots.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions, or \( x \)-values, where the function equals zero. They can be found using the quadratic formula once the discriminant has been evaluated.
Depending on the value of the discriminant, the roots may be:

• Real and distinct
• Real and equal
• Complex

For the equation \( s^2 + 10s + 8 = 0 \), with a discriminant of 68, the roots are real and distinct: \( s_1 = -3.87 \) and \( s_2 = -2.13 \). Finding the roots confirms the places where the graph of the quadratic equation intersects the \( x \)-axis.