M(x)=(3x-1)(3-x)+4x2+19

Solution for M(x)=(3x-1)(3-x)+4x2+19 equation:

(M)=(3M-1)(3-M)+4M^2+19

We move all terms to the left:

(M)-((3M-1)(3-M)+4M^2+19)=0

We add all the numbers together, and all the variables

-((3M-1)(-1M+3)+4M^2+19)+M=0

We multiply parentheses ..

-((-3M^2+9M+M-3)+4M^2+19)+M=0


We calculate terms in parentheses: -((-3M^2+9M+M-3)+4M^2+19), so:

(-3M^2+9M+M-3)+4M^2+19

We add all the numbers together, and all the variables

4M^2+(-3M^2+9M+M-3)+19

We get rid of parentheses

4M^2-3M^2+9M+M-3+19

We add all the numbers together, and all the variables

M^2+10M+16

Back to the equation:

-(M^2+10M+16)


We add all the numbers together, and all the variables

M-(M^2+10M+16)=0

We get rid of parentheses

-M^2+M-10M-16=0

We add all the numbers together, and all the variables

-1M^2-9M-16=0

a = -1; b = -9; c = -16;
Δ = b2-4ac
Δ = -92-4·(-1)·(-16)
Δ = 17
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$M_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$M_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$M_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-\sqrt{17}}{2*-1}=\frac{9-\sqrt{17}}{-2}
$
$M_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+\sqrt{17}}{2*-1}=\frac{9+\sqrt{17}}{-2}
$