Solve (4x-1)*(4x+1)-(4x+1-4)^2+(y-1)^2+8

$$( 4 x - 1 ) \cdot ( 4 x + 1 ) - ( 4 x + 1 - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + 8$$

Evaluate

$$\left(y-1\right)^{2}+24x-2$$

Solution Steps

Consider \left(4x-1\right)\left(4x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\left(4x\right)^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Expand \left(4x\right)^{2}.

$$4^{2}x^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Subtract 4 from 1 to get -3.

$$16x^{2}-1-\left(4x-3\right)^{2}+\left(y-1\right)^{2}+8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-3\right)^{2}.

$$16x^{2}-1-\left(16x^{2}-24x+9\right)+\left(y-1\right)^{2}+8$$

To find the opposite of 16x^{2}-24x+9, find the opposite of each term.

$$16x^{2}-1-16x^{2}+24x-9+\left(y-1\right)^{2}+8$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-1+24x-9+\left(y-1\right)^{2}+8$$

Subtract 9 from -1 to get -10.

$$-10+24x+\left(y-1\right)^{2}+8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-1\right)^{2}.

$$-10+24x+y^{2}-2y+1+8$$

Add -10 and 1 to get -9.

$$-9+24x+y^{2}-2y+8$$

Add -9 and 8 to get -1.

$$-1+24x+y^{2}-2y$$

Expand

$$24x+y^{2}-2y-1$$

Solution Steps

Consider \left(4x-1\right)\left(4x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\left(4x\right)^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Expand \left(4x\right)^{2}.

$$4^{2}x^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-1-\left(4x+1-4\right)^{2}+\left(y-1\right)^{2}+8$$

Subtract 4 from 1 to get -3.

$$16x^{2}-1-\left(4x-3\right)^{2}+\left(y-1\right)^{2}+8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-3\right)^{2}.

$$16x^{2}-1-\left(16x^{2}-24x+9\right)+\left(y-1\right)^{2}+8$$

To find the opposite of 16x^{2}-24x+9, find the opposite of each term.

$$16x^{2}-1-16x^{2}+24x-9+\left(y-1\right)^{2}+8$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-1+24x-9+\left(y-1\right)^{2}+8$$

Subtract 9 from -1 to get -10.

$$-10+24x+\left(y-1\right)^{2}+8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-1\right)^{2}.

$$-10+24x+y^{2}-2y+1+8$$

Add -10 and 1 to get -9.

$$-9+24x+y^{2}-2y+8$$

Add -9 and 8 to get -1.

$$-1+24x+y^{2}-2y$$