SOLVING QUADRATIC EQUATIONS – AI I .4B

Quadratic Formula

Discriminant

The discriminant tells you what type of roots your equation will have. This can help you decide the best/easiest way to solve it.

Quadratic Equation Standard Form: ax2 + bx + c = 0 a, b, and c are coefficients!

Discriminant: (b)2 – 4ac Remember, just type the whole thing into the

calculator at once!! Don’t forget the parentheses.

Discriminant

Value of the Discriminant Nature of the Solutions

Negative 2 imaginary solutions

Zero 1 Real – Rational Solution

Positive – perfect square 2 Real – Rational Solutions

Positive – non-perfect square

2 Real – Irrational Solutions

So how do I find those solutions??

Quadratic formula:

Wait, does something look familiar? Let’s rewrite it!

The ‘opposite’ of b. The – just changes the sign of b.

Quadratic Formula – the Steps

1) find the discriminant2) plug into the quadratic formula for –b, the

discriminant, and a. 3) simplify the radical and denominator4) simplify the fraction

Split the fraction into two if the solutions are rational Just reduce the fraction if the solutions are irrational

or imaginary

Let’s look at an example

Solve: 1) discriminant

Are we in the correct format? Set the equation equal to zero a = 3, b = 4, c = 6 Since our discriminant is negative, we have 2

imaginary solutions

Let’s look at an example

Solve: 1) discriminant: 2 imaginary solutions2) plug into the quadratic formula

3) simplify the radical and denominator =

Let’s look at an example

Solve: 1) discriminant: 2 imaginary solutions2) plug into the quadratic formula: 3) simplify the radical and denominator4) simplify the fraction

Since our solutions are imaginary, there is no need to split it.

Can I reduce my coefficients?? Yes, divide them by 2!

The solutions to are

x =

= =

x =

Solve:

1) Discriminant:

a = 4; b = -11; c = 6 (b)2 – 4ac => (-11)2 – 4(4)(6) = 25 2 real, rational roots

2) Quadratic Formula

Solve:

1) Discriminant: 25; 2 real, rational roots2) Quadratic Formula: x 3) Simplify the radical and denominator

4) Simplify the fraction

Since there are rational roots, split the fraction up! and = and and

Try some on your own…

2

2

2

2

2

1. 63 2

2. 36

3. 24 5

4. 7 13 0

5. 3 5 6 0

x x

x

x x

x x

x x

1. 9,7

2. 6

3. 3,8

7 34.

2

5 475.

6

x

x i

x

ix

ix