Solve 6(x+5)^2-5(x^2-y^2)-6(x-9)^2

$$6 ( x + 5 ) ^ { 2 } - 5 ( x ^ { 2 } - y ^ { 2 } ) - 6 ( x - 9 ) ^ { 2 }$$

Evaluate

$$-5x^{2}+168x+5y^{2}-336$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$6\left(x^{2}+10x+25\right)-5\left(x^{2}-y^{2}\right)-6\left(x-9\right)^{2}$$

Use the distributive property to multiply 6 by x^{2}+10x+25.

$$6x^{2}+60x+150-5\left(x^{2}-y^{2}\right)-6\left(x-9\right)^{2}$$

Use the distributive property to multiply -5 by x^{2}-y^{2}.

$$6x^{2}+60x+150-5x^{2}+5y^{2}-6\left(x-9\right)^{2}$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$x^{2}+60x+150+5y^{2}-6\left(x-9\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$x^{2}+60x+150+5y^{2}-6\left(x^{2}-18x+81\right)$$

Use the distributive property to multiply -6 by x^{2}-18x+81.

$$x^{2}+60x+150+5y^{2}-6x^{2}+108x-486$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+60x+150+5y^{2}+108x-486$$

Combine 60x and 108x to get 168x.

$$-5x^{2}+168x+150+5y^{2}-486$$

Subtract 486 from 150 to get -336.

$$-5x^{2}+168x-336+5y^{2}$$

Expand

$$-5x^{2}+168x+5y^{2}-336$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$6\left(x^{2}+10x+25\right)-5\left(x^{2}-y^{2}\right)-6\left(x-9\right)^{2}$$

Use the distributive property to multiply 6 by x^{2}+10x+25.

$$6x^{2}+60x+150-5\left(x^{2}-y^{2}\right)-6\left(x-9\right)^{2}$$

Use the distributive property to multiply -5 by x^{2}-y^{2}.

$$6x^{2}+60x+150-5x^{2}+5y^{2}-6\left(x-9\right)^{2}$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$x^{2}+60x+150+5y^{2}-6\left(x-9\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$x^{2}+60x+150+5y^{2}-6\left(x^{2}-18x+81\right)$$

Use the distributive property to multiply -6 by x^{2}-18x+81.

$$x^{2}+60x+150+5y^{2}-6x^{2}+108x-486$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+60x+150+5y^{2}+108x-486$$

Combine 60x and 108x to get 168x.

$$-5x^{2}+168x+150+5y^{2}-486$$

Subtract 486 from 150 to get -336.

$$-5x^{2}+168x-336+5y^{2}$$