# Find all real solutions of the equation by factoring. 6x(x-1)=21-x

Find all real solutions of the equation by factoring.
6x(x-1)=21-x
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Kirsten Davis
Step 1
We have to find all real solutions of the equation by factorizing:
6x(x-1)=21-x
Solving given equation,
6x(x-1)=21-x
$6{x}^{2}-6x=21-x$
$6{x}^{2}-6x+x=21$
$6{x}^{2}-5x-21=0$
Now solving by factorization,
$6{x}^{2}-14x+9x-21=0$
2x(3x-7)+3(3x-7)=0
(3x-7)(2x+3)=0
Step 2
Either,
3x-7=0
3x=7
$x=\frac{7}{3}$
or,
2x+3=0
2x=-3
$x=-\frac{3}{2}$
Hence, solutions of the equation are $x=-\frac{3}{2},\frac{7}{3}$.

Hector Roberts
The given quadratic equation is 6x(x-1)=21-x
6x(x-1)=21-x [given equation]
$6{x}^{2}-6x=21-x$
$6{x}^{2}-6x+x=21-x+x$
$6{x}^{2}-5x=21$
$6{x}^{2}-5x-21=21-21$
$6{x}^{2}-5x-21=0$ [quadratic form]
$6{x}^{2}+9x-14x-21=0$
3x(2x+3)-7(2x+3)=0
(3x-7)(2x+3)=0 [factor]
3x-7=0 or 3x+3=0 [zero-product property]
$x=\frac{7}{3}$ or $x=\frac{-3}{2}$
The solutions are $x=\frac{7}{3}$ and $x=\frac{-3}{2}$