How do I factor this trinomial that has a radical ?

Question

x² +3√2 x —5=0 

 

* the X is not part of the 2 *

Answer 1

Alex,

 

To factor a quadratic trinomial in which the leading coefficient is 1, you must find two numbers a and b, whose product is −5 and whose sum is 3√2. One way to do this would be to solve the system of equations

 

ab = −5

a + b = 3√2

 

This would eventually lead to the original quadratic equation in some form or another.

 

Another way to do this would be to simply use the quadratic formula to solve the equation for its two roots, r₁ and r₂, then form the two factors (x - r₁)(x - r₂). This would perhaps be more straightforward.

 

I will leave the rest to you, but you are free to reply back if you still need help with this. I will tell you that the roots work out rather nicely and both involve the square root of 2.

Answer 2

The discriminant of the quadratic equation, b²-4ac, equals 38 in this case.

Therefore the solutions of this equation are ½[-3√(2)±√(38)] and this gives us two roots, r₂ and r₂.

 

The assignment to factor the equation would be messy if the exact surd values were substituted into (x-r₁)(x-r₂) = 0.

 

Awkward 'problem'--abnormal, even. 

Answer 3

 

 complete the square

x^2+3√2x=5

(x+3√2/2)^2=19/2

(x+3√2/2)^2=(√(19/2))^2

(x+3√2/2-√19/√2)(x+3√2/2+√19/√2)=0

(x+(3-√19)/√2)(x+(3+√19)/√2)=0