Solve left(x^2-6right)^2+13(x^2-6x)+40

$${ \left( { x }^{ 2 } -6 \right) }^{ 2 } +13( { x }^{ 2 } -6x)+40$$

Evaluate

$$x^{4}+x^{2}-78x+76$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-6\right)^{2}.

$$\left(x^{2}\right)^{2}-12x^{2}+36+13\left(x^{2}-6x\right)+40$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-12x^{2}+36+13\left(x^{2}-6x\right)+40$$

Use the distributive property to multiply 13 by x^{2}-6x.

$$x^{4}-12x^{2}+36+13x^{2}-78x+40$$

Combine -12x^{2} and 13x^{2} to get x^{2}.

$$x^{4}+x^{2}+36-78x+40$$

Add 36 and 40 to get 76.

$$x^{4}+x^{2}+76-78x$$

Expand

$$x^{4}+x^{2}-78x+76$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-6\right)^{2}.

$$\left(x^{2}\right)^{2}-12x^{2}+36+13\left(x^{2}-6x\right)+40$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-12x^{2}+36+13\left(x^{2}-6x\right)+40$$

Use the distributive property to multiply 13 by x^{2}-6x.

$$x^{4}-12x^{2}+36+13x^{2}-78x+40$$

Combine -12x^{2} and 13x^{2} to get x^{2}.

$$x^{4}+x^{2}+36-78x+40$$

Add 36 and 40 to get 76.

$$x^{4}+x^{2}+76-78x$$