Completing the square ... again.

To make x2 + bx a perfect square, for example x2 − 6x,

1.

2.

A quadratic equation is an equation of the form

The solutions to a quadratic equation (and any polynomial equation
for that matter) are called .

Example. Solve the following quadratic equation:

• x2 − 2x − 3 = 0

• x2 − 2x − 4 = 0

Example. Solve the equation 3x2 + 6bx + 4 = 0 (in terms of b)

Example. Solve the equation ax2 +bx+c = 0 (in terms of a, b, c).

Thus, the quadratic formula says that the solutions to ax2+bx+c = 0
are

Example. Solve the following quadratic equations:

• x2 − 12x + 35 = 0

• x2 − 12x + 36 = 0

• x2 − 12x + 37 = 0

Therefore, the number of solutions you get to a quadratic equation
is either

The product and sum of roots.

Suppose the roots of x 2 + bx + c = 0 are and .

Then x2 + bx + c =

So

Example. Find the roots of the following equations:

 • x2 − 2x − 3 = 0 • x2 + x − 1 = 0 Product of roots : Product of roots : Sum of roots : Sum of roots :

Example. Find the sum and product of the roots of the following
equations :

• x2 + 4x − 7 = 0

• 2x2 + 6x − 135 = 0

The Discriminant.

The discriminant of the quadratic equation ax2 + bx + c = 0 is

The discriminant can be used to tell how many solutions a quadratic
equation will have.

• If , then there are solution (s).

• If , then there are solution (s).

• If , then there are solution(s).

Example. Find the number of real solutions for 2x2 − 3x − 1 = 0.

Example. Find a value of k which makes kx2 + 4x − 7 = 0 have
no real solutions .

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