Given that  α and β are the roots of the equation 2x2+5x−1=0 ,find a quadratic equation with roots α+2  and β+2 .Please give 1 answer.2x2−3x−3=0 −2x2−3x−3=0 2x2+3x−3=0 2x2+3x+3=0

The roots of the given equation 2x^2 + 5x - 1 = 0 are α and β.

We are asked to find a quadratic equation with roots α+2 and β+2.

The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a and the product of the roots is given by c/a.

So, the sum of the roots α+2 and β+2 is (α+2) + (β+2) = α + β + 4.

The product of the roots α+2 and β+2 is (α+2)(β+2) = αβ + 2α + 2β + 4.

We know that α + β = -b/a = -5/2 and αβ = c/a = -1/2 from the given equation.

Substituting these values in, we get:

Sum of the roots = α + β + 4 = -5/2 + 4 = 3/2.

Product of the roots = αβ + 2α + 2β + 4 = -1/2 + 2(-5/2) + 4 = -3.

Therefore, the quadratic equation with roots α+2 and β+2 is given by:

x^2 - (sum of roots)x + product of roots = 0

=> x^2 - (3/2)x - 3 = 0

Multiplying through by 2 to clear the fraction gives:

2x^2 - 3x - 6 = 0

So, the quadratic equation with roots α+2 and β+2 is 2x^2 - 3x - 6 = 0.