Question

Find the quadratic equation whose solutions have a sum of 3/4 and a product of 1/8. To start, you will need to find the values of the coefficient a, b, and c. Then show that equation works by solving the equation, followed by checking that the solutions have the indicated sum and product. Your final equation should have coefficients that are integers, with no common factors between them all (other than 1). Please please please help me!!!

Answer 1

The equation is
`8x^2 -6x+1=0`

Explanation:

We need to find the quadratic equation which is in the form:

`ax^2 + bx + c = 0`

We are given sum S = 3/4 and Product P = 1/8

The quadratic equation in terms of sum and products can be written as:

`x^2 - Sx + P =0`

Where S is sum and P is product. Putting their values we get:

`x^2-(3)/(4)x+(1)/(8)=0`

The co-efficient should be integer so, taking LCM of 4 and 8

`(8x^2 -6x+1)/(8) =0\\Dividing\,\,both\,\,sides\,\,by\,\,8\\8x^2 -6x+1=0`

So,

Co-efficient are: a = 8 , b = -6 and c= 1

Solving the equation:

`8x^2 -6x+1=0\\8x^2 -2x -4x+1=0\\8x(x-1/4)-4(x-1/4) =0\\(x-1/4)(8x-4)=0\\x-1/4 =0\,\, and\,\, 8x-4 =0\\x = 1/4\,\, and x \,\,= 4/8 = 1/2`

So values of x are 1/4 and 1/2

Sum: 1/4+1/2 = 1+2/4 = 3/4

Product: 1/4 * 1/2 = 1/8