Quadratics can be solved by equating one side of the equation to zero and factorising. For a product to be equal to zero, one or both of the factors must be equal to zero. Therefore, the quadratic equation can be solved by equating each factor to zero and solving for the variable.

Quadratic equations that cannot be factorised can be solved by completing the square. Completing the square results in a square plus a term, which can be solved by moving the additional term to the other side of the equation and square rooting both sides.

A common method to solve quadratics is by using the quadratic formula. Substituting coefficients a, b, c of the quadratic equation into the formula gives its solutions.

Rearranging gives x² + 15x + 36 = 0.
Factorising gives (x + 12)(x + 3) = 0.
Equating each factor to 0 gives x + 12 = 0 or x + 3 = 0.
Thus, x = −12 or x = −3.

x = {−12, −3}

The perimeter of the rectangle is given by x² + x² + 3 − 2x + 3 − 2x = 2x² − 4x + 6.
Since the perimeter is 6, 2x² − 4x + 6 = 6.
Rearranging gives 2x² − 4x = 0.
Factorising gives 2x(x − 2) = 0.
Equating each factor to 0 gives 2x = 0 or x − 2 = 0.
Thus, x = 0 or x = 2.
When x = 0, two sides of the rectangle would be 0, which is impossible.
When x = 2, one of the sides of the rectangle is less than 0, which is impossible.
Therefore, there are no solutions.

no solutions

x = (−3 ± √(3² − 4 × 1 × (−4)))/(2 × 1).
x = (−3 ± 5)/2

x = {−4, 1}

area of rectangle = (2x + 11)(4x − 1) = 8x² − 2x + 44x − 11 = 8x² + 42x − 11
8x² + 42x − 11 = 20
8x² + 42x − 31 = 0
x = (−42 ± √(42² − 4(8)(−31)))/2(8)
x = (−42 ± 2√689)/16 = (−21 ± √689)/8
x = (−21 − √689)/8 would make two of the sides of the rectangle negative, which is impossible, therefore it is rejected. Thus, x = (−21 + √689)/8 = 0.656 (to 3 significant figures).

0.656

Factorising gives (x + 2)(x + 1) = 0.
Equating each factor to 0 gives x + 2 = 0 or x + 1 = 0.
Thus, x = −2 or x = −1.

x = {−2, −1}