# Quadratic Equations

** Completing the square ... again.**

To make x^{2} + bx a perfect square, for example x^{2} − 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:

• x^{2} − 2x − 3 = 0

• x^{2} − 2x − 4 = 0

**The Quadratic Formula .**

**Example. **Solve the equation 3x^{2} + 6bx + 4 = 0 (in terms of b)

**The Quadratic Formula **

**Example.** Solve the equation ax^{2} +bx+c = 0 (in terms of a, b, c).

Thus, the quadratic formula says that the solutions to
ax^{2}+bx+c = 0

are

**Example.** Solve the following quadratic equations:

• x^{2} − 12x + 35 = 0

• x^{2} − 12x + 36 = 0

• x^{2} − 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 x^{2} + bx + c =

So

**Example. **Find the roots of the following equations:

• x^{2} − 2x − 3 = 0 |
• x^{2} + 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 :

• x^{2} + 4x − 7 = 0

• 2x^{2} + 6x − 135 = 0

**The Discriminant.**

The **discriminant **of the quadratic equation ax^{2} + 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 2x^{2} − 3x − 1 = 0.

**Example.** Find a value of k which makes kx^{2} + 4x − 7 = 0
have

no real solutions .

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