Find the quadratic polynomial whose sum of zeroes is 1 4 and the product of the zeroes is - 1 .

We can find the quadratic equation if we know the sum of the roots and product of the roots. We know that if α and β are the zeroes of a quadratic polynomial p x , then, the polynomial p ( x ) can be written as p x = a x 2 - α + β x + α β , where a is a non-zero real number. Given, sum of the roots = α + β = 1 4 and product of the roots = α β = - 1 Hence, the quadratic polynomial p x can be written as p x = a x 2 - 1 4 x - 1 = a 4 x 2 - x - 4 4 By taking a = 4 , we get one of the quadratic polynomials which satisfy the given conditions. Hence, the quadratic polynomial is 4 x 2 - x - 4 .

Find the quadratic polynomial whose sum of zeroes is 2 and the product of the zeroes is 1 3 .

We can find the quadratic equation if we know the sum of the roots and product of the roots. We know that if α and β are the zeroes of a quadratic polynomial p x , then, the polynomial p ( x ) can be written as p x = a x 2 - α + β x + α β , where a is a non-zero real number. Given, sum of the roots = α + β = 2 and product of the roots = α β = 1 3 Hence, the quadratic polynomial p x can be written as p x = a x 2 - 2 x - 1 3 = a 3 x 2 - 3 2 x + 1 3 By taking a = 3 , we get one of the quadratic polynomials which satisfy the given conditions. Hence, the quadratic polynomial is 3 x 2 - 3 2 x + 1 .

Find the quadratic polynomial whose sum of zeroes is - 1 4 and the product of the zeroes is 1 4 .

We can find the quadratic equation if we know the sum of the roots and product of the roots. We know that if α and β are the zeroes of a quadratic polynomial p x , then, the polynomial p ( x ) can be written as p x = a x 2 - α + β x + α β , where a is a non-zero real number. Given, sum of the roots = α + β = - 1 4 and product of the roots = α β = 1 4 Hence, the quadratic polynomial p x can be written as p x = a x 2 - - 1 4 x + 1 4 = a 4 x 2 + x + 1 4 By taking a = 4 , we get one of the quadratic polynomials which satisfy the given conditions. Hence, the quadratic polynomial is 4 x 2 + x + 1 .