Question

1.
What is the sum of the two solutions to the equation 3x2 - x = 4?

Answer 1

The sum of the two solutions to the given equation is 1/3.

Explanation:

Nature of Roots of a Second-Degree Equation

Given a quadratic equation:

`ax^2 + bx + c = 0`

The roots of the equation are:

`\displaystyle x_1=(-b+ √(b^2-4ac))/(2a)`

`\displaystyle x_2=(-b- √(b^2-4ac))/(2a)`

The sum of the roots is:

`\displaystyle x_1+x_2=-(b)/(a)`

The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient.

We are given the equation:

`3x^2 - x = 4`

Rearranging:

`3x^2 - x - 4=0`

The coefficient b is b=-1, and the leading coefficient is a=3, thus the sum of the roots is:

`\displaystyle x_1+x_2=-(-1)/(3)`

`\displaystyle x_1+x_2=(1)/(3)`

The sum of the two solutions to the given equation is 1/3.