Solve (x+sqrt{2})(x+3sqrt{2})

$$( x + \sqrt { 2 } ) ( x + 3 \sqrt { 2 } )$$

Evaluate

$$\left(x+\sqrt{2}\right)\left(x+3\sqrt{2}\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x+\sqrt{2} by each term of x+3\sqrt{2}.

$$x^{2}+3x\sqrt{2}+\sqrt{2}x+3\left(\sqrt{2}\right)^{2}$$

Combine 3x\sqrt{2} and \sqrt{2}x to get 4x\sqrt{2}.

$$x^{2}+4x\sqrt{2}+3\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$x^{2}+4x\sqrt{2}+3\times 2$$

Multiply 3 and 2 to get 6.

$$x^{2}+4x\sqrt{2}+6$$

Differentiate w.r.t. x

$$2\left(x+2\sqrt{2}\right)$$

Steps Using Derivative Rule for Sum

Apply the distributive property by multiplying each term of x+\sqrt{2} by each term of x+3\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+3x\sqrt{2}+\sqrt{2}x+3\left(\sqrt{2}\right)^{2})$$

Combine 3x\sqrt{2} and \sqrt{2}x to get 4x\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+4x\sqrt{2}+3\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+4x\sqrt{2}+3\times 2)$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+4x\sqrt{2}+6)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2x^{2-1}+4\sqrt{2}x^{1-1}$$

Subtract 1 from 2.

$$2x^{1}+4\sqrt{2}x^{1-1}$$

Subtract 1 from 1.

$$2x^{1}+4\sqrt{2}x^{0}$$

For any term t, t^{1}=t.

$$2x+4\sqrt{2}x^{0}$$

For any term t except 0, t^{0}=1.

$$2x+4\sqrt{2}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$2x+4\sqrt{2}$$