Zeroes of Quadratic Polynomials

To find the zeroes of a quadratic polynomial, we need to solve the equation when the polynomial is equal to zero. The zeroes of a quadratic polynomial are the values of x at which the polynomial equals zero.

Quadratic Polynomial 1: 5x^2 - 4 - 8x

To find the zeroes of the quadratic polynomial 5x^2 - 4 - 8x, we set the polynomial equal to zero and solve for x:

5x^2 - 4 - 8x = 0

Step 1: Move all the terms to one side of the equation:

5x^2 - 8x - 4 = 0

Step 2: Factorize the quadratic expression:

(5x + 2)(x - 2) = 0

Step 3: Set each factor equal to zero and solve for x:

5x + 2 = 0 or x - 2 = 0

Solving the first equation:

5x = -2

x = -2/5

Solving the second equation:

x = 2

Therefore, the zeroes of the quadratic polynomial 5x^2 - 4 - 8x are x = -2/5 and x = 2.

Quadratic Polynomial 2: 3x^2 - 4x - 4

To find the zeroes of the quadratic polynomial 3x^2 - 4x - 4, we set the polynomial equal to zero and solve for x:

3x^2 - 4x - 4 = 0

Step 1: Move all the terms to one side of the equation:

3x^2 - 4x - 4 = 0

Step 2: Factorize the quadratic expression:

(3x + 2)(x - 2) = 0

Step 3: Set each factor equal to zero and solve for x:

3x + 2 = 0 or x - 2 = 0

Solving the first equation:

3x = -2

x = -2/3

Solving the second equation:

x = 2

Therefore, the zeroes of the quadratic polynomial 3x^2 - 4x - 4 are x = -2/3 and x = 2.

Quadratic Polynomial 3: x^2 + 7x + 12

To find the zeroes of the quadratic polynomial x^2 + 7x + 12, we set the polynomial equal to zero and solve for x:

x^2 + 7x + 12 = 0

Step 1: Move all the terms to one side of the equation:

x^2 + 7x + 12 = 0

Step 2: Factorize the quadratic expression:

(x + 3)(x + 4) = 0

Step 3: Set each factor equal to zero and solve for x:

x + 3 = 0 or x + 4 = 0

Solving the first equation:

x = -3