confirm that (2x^2 + 1)^2 + (2x^2 + 2x)^2 = (2x^2 + 2x + 1)^2
Answers
refer the attachment for the answer
2x
2
−2x+1
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x
2
+Bx+C=0.This is achieved by dividing both sides of the equation by 2
x
2
−1x+
2
1
=0
Let r and s be the factors for the quadratic equation such that x
2
+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs=C
r+s=1
rs=
2
1
Two numbers r and s sum up to 1 exactly when the average of the two numbers is
2
1
∗1=
2
1
. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x
2
+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
r=
2
1
−u
s=
2
1
+u
To solve for unknown quantity u, substitute these in the product equation rs=
2
1
(
2
1
−u)(
2
1
+u)=
2
1
Simplify by expanding (a−b)(a+b)=a
2
–b
2
4
1
−u
2
=
2
1
Simplify the expression by subtracting
4
1
on both sides
−u
2
=
2
1
−
4
1
=
4
1
Simplify the expression by multiplying −1 on both sides and take the square root to obtain the value of unknown variable u
u
2
=−
4
1
u=±
−
4
1
=±
2
1
i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
r=
2
1
−
2
1
i=0.500−0.500i
s=
2
1
+
2
1
i=0.500+0.500i