write a quadratic equation in ax^2 + bx + c = 0 form

Question

one root:

-2 + 3 square root of 2

Answer 1

When we use the quadratic formula to find the roots, we will have 2 solution because of the ± sign in the formula.

 

If one root is -2 + 3√(2)

 

then the other root is -2 - 3√(2)

 

 

Remember that when we find the roots, we set the factors equal to zero. 

 

(x + 2 - 3√2)(x + 2 + 3√2) = 0

 

We can expand this out.  The easiest way to do this is to use distributive property by utilizing one of the factors as the GCF. 

 

x(x + 2 + 3√2) + 2(x + 2 + 3√2) - 3√2(x + 2 + 3√2) = 0

 

x² + 2x + 3x√(2) + 2x + 4 + 6√(2) - 3x√(2) - 6√(2) - 18 = 0

 

 

Next is to simplify by combining like terms.

 

x² + 4x -14  = 0

 

 

The quadratic equation is

 

f(x) = x² + 4x - 14

Answer 2

The roots are -2+3√2 and -2-3√2

 

^{Sum of roots = -b/a  = -4}

 

^{Product of roots = c/a = -14}

 

^{So, if we let a = 1, then b = 4 and c = -14}

 

x² + 4x -14 has the given root.