Solve beta=(x-2)(x-3)+2(x^2-4)-(x-2)^2

$$\beta = ( x - 2 ) ( x - 3 ) + 2 ( x ^ { 2 } - 4 ) - ( x - 2 ) ^ { 2 }$$

Solve for β

$$\beta =-\left(6+x-2x^{2}\right)$$

Solution Steps

Use the distributive property to multiply x-2 by x-3 and combine like terms.

$$\beta =x^{2}-5x+6+2\left(x^{2}-4\right)-\left(x-2\right)^{2}$$

Use the distributive property to multiply 2 by x^{2}-4.

$$\beta =x^{2}-5x+6+2x^{2}-8-\left(x-2\right)^{2}$$

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$\beta =3x^{2}-5x+6-8-\left(x-2\right)^{2}$$

Subtract 8 from 6 to get -2.

$$\beta =3x^{2}-5x-2-\left(x-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$\beta =3x^{2}-5x-2-\left(x^{2}-4x+4\right)$$

To find the opposite of x^{2}-4x+4, find the opposite of each term.

$$\beta =3x^{2}-5x-2-x^{2}+4x-4$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$\beta =2x^{2}-5x-2+4x-4$$

Combine -5x and 4x to get -x.

$$\beta =2x^{2}-x-2-4$$

Subtract 4 from -2 to get -6.

$$\beta =2x^{2}-x-6$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{8\beta +49}+1}{4}$$

$$x=\frac{\sqrt{8\beta +49}+1}{4}$$

Solve for x

$$x=\frac{-\sqrt{8\beta +49}+1}{4}$$

$$x=\frac{\sqrt{8\beta +49}+1}{4}\text{, }\beta \geq -\frac{49}{8}$$