Quadratic Formula (C)

A quadratic equation is an equation of the form

 

Az² + Bz + C = 0

where $A\neq 0$, B, C and z are complex numbers. We regard z as unknown, and A, B, and C as known. The ojective is to determine z. Completing the square gives us

Theorem 2622 (Quadratic Formula)

If $A\neq 0$, B and C are complex numbers and z satisfies Az² + Bz + C = 0 then

$$z = \frac{-B - \sqrt{B^2-4AC}}{2A}\;\;{\rm or}\;\; z = \frac{-B + \sqrt{B^2-4AC}}{2A}$$

where $(\sqrt{B^2-4AC})^2 = B^2 - 4AC$.

Note that we have to be careful about the meaning of $\sqrt{B^2-4AC}$ when B² - 4AC is not a non-negative real number. If B is a real number and $AC \leq 0$, then the discriminant B² - 4AC is a non-negative real number, and so is its square root. If either B or C is zero, it is more efficient to solve quadratic equations by factoring.

Exercises

1.

Solve each equation for the indicated variable:

(a)

z² + 3z + 2 = 0, solve for z.

(b)

x² + 7x + 2 = 0, solve for x.

(c)

x² + 7x - 2 = 0, solve for x.

(d)

x² + 4x + 5 = 0, solve for x.

(e)

2x²-9x + 11 = 0, solve for x.

(f)

(2+i)x² + 20x + (2-i) = 0, solve for x.

(g)

(2+3i)x² +2ix + (2-3i) = 0, solve for x.

(h)

2z² + iz + 11 = 0, solve for z.

(i)

2x² + 4xy + y² + 2y + 3x -12 = 0, solve for y.

(j)

16x² + 4xy + y² + 2y + 3x -12 = 0, solve for y.

(k)

5x² + 4xy + 4y² + 2y + 3x -12 = 0, solve for y.

2.

For which real numbers x are there real numbers y so that 2x² + 4xy + y² + 2y + 3x -12 = 0?

3.

Derive the quadratic formula by completing the square.