Solve 15x^2+8xsqrt{45}+48

$$15 { x }^{ 2 } +8x \sqrt{ 45 } +48$$

Evaluate

$$15x^{2}+24\sqrt{5}x+48$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$15x^{2}+8x\times 3\sqrt{5}+48$$

Multiply 8 and 3 to get 24.

$$15x^{2}+24x\sqrt{5}+48$$

Differentiate w.r.t. x

$$30x+24\sqrt{5}$$

Steps Using Derivative Rule for Sum

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}.

$$2\times 15x^{2-1}+8\sqrt{45}x^{1-1}$$

Multiply 2 times 15.

$$30x^{2-1}+8\sqrt{45}x^{1-1}$$

Subtract 1 from 2.

$$30x^{1}+8\sqrt{45}x^{1-1}$$

Subtract 1 from 1.

$$30x^{1}+24\sqrt{5}x^{0}$$

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

$$30x+24\sqrt{5}x^{0}$$

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

$$30x+24\sqrt{5}\times 1$$

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

$$30x+24\sqrt{5}$$