How to prove that i don't need to find a discriminant to solve the quadratic y^2+6y+9=0?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The solutions of the quadratic equation y^2 + 6y + 9 = 0 can easily be found without finding the discriminant.

y^2 + 6y + 9 = 0

=> y^2 + 3y + 3y + 9 = 0

=> y( y + 3) + 3(y + 3) = 0

=> (y + 3)^2 = 0

=> y = -3

Or you can use the formula (a + b)^2 = a^2 + b^2 + 2ab.

The solution of the quadratic equation obtained without finding the discriminant is -3

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Well, it is simple, since we recognize in the given form of the quadratic a perfect square.

We've had been driven by the formula:

(a+b)^2 = a^2 + 2ab + b^2

If we'll put a^2 = y^2 and b^2 = 9 => b=3 and a=y

2ab = 2*3*y = 6y

y^2+6y+9 = (y+3)^2

We'll solve the quadratic:

(y+3)^2 = 0

(y+3)(y+3) = 0

We'll put y+3=0

y=-3

The solutions of the quadratic are real numbers and they are equal: y1=y2 = -3.

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