Can we determine a quadratic equation if we

have its roots?

Do Now: Use the quadratic formula to determine the general form of BOTH

roots to any quadratic equation.

What is the sum of the roots of a quadratic equation?

• We determined that the general form of the two roots can be written as:

• To find the sum, we add these together

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x =−b+ b2 − 4ac

2a or x =

−b− b2 − 4ac

2a

€

−b+ b2 − 4ac

2a+

−b− b2 − 4ac

2a

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=−b+ b2 − 4ac −b+ b2 − 4ac

2a

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=−b−b

2a=

−2b

2a=

−b

a

The sum of the roots is always the same?

• Yes, for any quadratic equation, due to the nature of the roots, the sum of the roots is the opposite of b over a.

• If we know one root and the sum of the roots, we can find the other root.

• Note: If we know that one root is imaginary, then the other root is the CONJUGATE!!!

Is there a similar relationship for the product

of the roots?• Yes! We can use the general form of

the roots to find the product.

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−b+ b2 − 4ac

2a•

−b− b2 − 4ac

2a

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=(−b+ b2 − 4ac )(−b− b2 − 4ac )

2a • 2a

€

=b2 + b b2 − 4ac −b b2 − 4ac − ( b2 − 4ac )2

4a2

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=b2 − (b2 − 4ac)

4a2

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=4ac

4a2=c

a

Example• What are the sum and product of the

roots of the equation 3x2-6x+8=0

• Sum

• Product

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−ba

=−(−6)

3

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=6

3= 2

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c

a=

8

3

Why do we care about the sum and product of roots?

• If we know the sum and product, we can write the original quadratic equation.

• The sum is made of b and a, and the product is made of c and a, so we have everything we need to write the quadratic equation.

Example• Find the quadratic equation whose roots

are:

€

3 + 2 and 3 − 21) Find sum and

product

2) Find a, b, and c

3) Write equation

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(3+ 2) + (3− 2)

€

=6

€

(3 + 2)(3− 2)

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=9 − 3 2 + 3 2 − 4

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=9 − 2 = 7

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6 =−b

a⇒ b = −6, a =1

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7 =c

a⇒ c = 7

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0 = x 2 − 6x + 7

Try on your own• Find the quadratic equation whose roots

are 5+2i and 5-2i.

Summary/HW• How can we determine a quadratic

equation if we have the roots of the equation?

• HW pg 87, 1-10