# Given the roots 6 and 7, determine the quadratic equation?

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The roots of a quadratic equation are given as 6 and 7. This means that x - 6 = 0 and x - 7 = 0

We can write ( x - 6)( x - 7) = 0

=> x^2 - 6x - 7x + 42 = 0

=> x^2 - 13x + 42 = 0

Therefore the required quadratic equation is

x^2 - 13x + 42 = 0

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To determine the quadratic equation, we'll use Viete's relations:

x1 + x2 = S, where x1 = 6 and x2 = 7

x1 + x2 = 6+7

S = 13

x1*x2 = P

P = 6*7

P = 42

x^2 - Sx + P = 0

x^2 - 13x + 42 = 0

x^2 - 13x + 42 = 0

Another method of creating the quadratic when knowing the roots is to write the equation as a product of linear factors;

(x-x1)(x-x2) = 0

(x-6)(x-7) = 0

We'll remove the brackets:

x^2 -6x - 7x +42 = 0

We'll combine like terms and we'll get the quadratic equation:

x^2 - 13x + 42 = 0

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(x - 6) (x - 7)

Using FOIL, expand the equation

FOIL = First Outside Inside Last

x^2 - 7x - 6x + 42

Combine like terms

x^2-13x+42

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Given the roots 6 and 7, determine the quadratic equation?

saying the roots are 6 and 7 give you a clue just put them back in parentheses:

(x - 6) (x - 7)

now FOIL

x^2 - 7x - 6x + 42

combine:

x^2 - 13x + 42