Solve x^2=(4+sqrt{15})^2-2*4sqrt{15}*(1+frac{sqrt{15}}{4})

$$x ^ { 2 } = ( 4 + \sqrt { 15 } ) ^ { 2 } - 2 \times 4 \sqrt { 15 } \times ( 1 + \frac { \sqrt { 15 } } { 4 } )$$

Solve for x

$$x=1$$

$$x=-1$$

Steps Using Difference of Squares

Multiply both sides of the equation by 4.

$$4x^{2}=4\left(4+\sqrt{15}\right)^{2}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{15}\right)^{2}.

$$4x^{2}=4\left(16+8\sqrt{15}+\left(\sqrt{15}\right)^{2}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

The square of \sqrt{15} is 15.

$$4x^{2}=4\left(16+8\sqrt{15}+15\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Add 16 and 15 to get 31.

$$4x^{2}=4\left(31+8\sqrt{15}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use the distributive property to multiply 4 by 31+8\sqrt{15}.

$$4x^{2}=124+32\sqrt{15}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply 8 and 4 to get 32.

$$4x^{2}=124+32\sqrt{15}-32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Add 32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right) to both sides.

$$4x^{2}+32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)=124+32\sqrt{15}$$

Use the distributive property to multiply 32\sqrt{15} by 1+\frac{\sqrt{15}}{4}.

$$4x^{2}+32\sqrt{15}+32\sqrt{15}\times \frac{\sqrt{15}}{4}=124+32\sqrt{15}$$

Cancel out 4, the greatest common factor in 32 and 4.

$$4x^{2}+32\sqrt{15}+8\sqrt{15}\sqrt{15}=124+32\sqrt{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$4x^{2}+32\sqrt{15}+8\times 15=124+32\sqrt{15}$$

Multiply 8 and 15 to get 120.

$$4x^{2}+32\sqrt{15}+120=124+32\sqrt{15}$$

Subtract 124 from both sides.

$$4x^{2}+32\sqrt{15}+120-124=32\sqrt{15}$$

Subtract 124 from 120 to get -4.

$$4x^{2}+32\sqrt{15}-4=32\sqrt{15}$$

Subtract 32\sqrt{15} from both sides.

$$4x^{2}+32\sqrt{15}-4-32\sqrt{15}=0$$

Combine 32\sqrt{15} and -32\sqrt{15} to get 0.

$$4x^{2}-4=0$$

Divide both sides by 4.

$$x^{2}-1=0$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)=0$$

To find equation solutions, solve x-1=0 and x+1=0.

$$x=1 x=-1$$

Steps by Finding Square Root

Multiply both sides of the equation by 4.

$$4x^{2}=4\left(4+\sqrt{15}\right)^{2}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{15}\right)^{2}.

$$4x^{2}=4\left(16+8\sqrt{15}+\left(\sqrt{15}\right)^{2}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

The square of \sqrt{15} is 15.

$$4x^{2}=4\left(16+8\sqrt{15}+15\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Add 16 and 15 to get 31.

$$4x^{2}=4\left(31+8\sqrt{15}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use the distributive property to multiply 4 by 31+8\sqrt{15}.

$$4x^{2}=124+32\sqrt{15}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply 8 and 4 to get 32.

$$4x^{2}=124+32\sqrt{15}-32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply both sides of the equation by 4.

$$16x^{2}=4\left(124+32\sqrt{15}\right)-128\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply both sides of the equation by 4.

$$64x^{2}=16\left(124+32\sqrt{15}\right)-4\times 128\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use the distributive property to multiply 16 by 124+32\sqrt{15}.

$$64x^{2}=1984+512\sqrt{15}-4\times 128\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply -4 and 128 to get -512.

$$64x^{2}=1984+512\sqrt{15}-512\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use the distributive property to multiply -512\sqrt{15} by 1+\frac{\sqrt{15}}{4}.

$$64x^{2}=1984+512\sqrt{15}-512\sqrt{15}-512\sqrt{15}\times \frac{\sqrt{15}}{4}$$

Cancel out 4, the greatest common factor in 512 and 4.

$$64x^{2}=1984+512\sqrt{15}-512\sqrt{15}-128\sqrt{15}\sqrt{15}$$

Combine 512\sqrt{15} and -512\sqrt{15} to get 0.

$$64x^{2}=1984-128\sqrt{15}\sqrt{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$64x^{2}=1984-128\times 15$$

Multiply -128 and 15 to get -1920.

$$64x^{2}=1984-1920$$

Subtract 1920 from 1984 to get 64.

$$64x^{2}=64$$

Divide both sides by 64.

$$x^{2}=\frac{64}{64}$$

Divide 64 by 64 to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$4x^{2}=4\left(4+\sqrt{15}\right)^{2}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{15}\right)^{2}.

$$4x^{2}=4\left(16+8\sqrt{15}+\left(\sqrt{15}\right)^{2}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

The square of \sqrt{15} is 15.

$$4x^{2}=4\left(16+8\sqrt{15}+15\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Add 16 and 15 to get 31.

$$4x^{2}=4\left(31+8\sqrt{15}\right)-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Use the distributive property to multiply 4 by 31+8\sqrt{15}.

$$4x^{2}=124+32\sqrt{15}-8\times 4\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Multiply 8 and 4 to get 32.

$$4x^{2}=124+32\sqrt{15}-32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)$$

Add 32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right) to both sides.

$$4x^{2}+32\sqrt{15}\left(1+\frac{\sqrt{15}}{4}\right)=124+32\sqrt{15}$$

Use the distributive property to multiply 32\sqrt{15} by 1+\frac{\sqrt{15}}{4}.

$$4x^{2}+32\sqrt{15}+32\sqrt{15}\times \frac{\sqrt{15}}{4}=124+32\sqrt{15}$$

Cancel out 4, the greatest common factor in 32 and 4.

$$4x^{2}+32\sqrt{15}+8\sqrt{15}\sqrt{15}=124+32\sqrt{15}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$4x^{2}+32\sqrt{15}+8\times 15=124+32\sqrt{15}$$

Multiply 8 and 15 to get 120.

$$4x^{2}+32\sqrt{15}+120=124+32\sqrt{15}$$

Subtract 124 from both sides.

$$4x^{2}+32\sqrt{15}+120-124=32\sqrt{15}$$

Subtract 124 from 120 to get -4.

$$4x^{2}+32\sqrt{15}-4=32\sqrt{15}$$

Subtract 32\sqrt{15} from both sides.

$$4x^{2}+32\sqrt{15}-4-32\sqrt{15}=0$$

Combine 32\sqrt{15} and -32\sqrt{15} to get 0.

$$4x^{2}-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-4\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-4\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-4\right)}}{2\times 4}$$

Multiply -16 times -4.

$$x=\frac{0±\sqrt{64}}{2\times 4}$$

Take the square root of 64.

$$x=\frac{0±8}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±8}{8}$$

Now solve the equation x=\frac{0±8}{8} when ± is plus. Divide 8 by 8.

$$x=1$$

Now solve the equation x=\frac{0±8}{8} when ± is minus. Divide -8 by 8.

$$x=-1$$

The equation is now solved.

$$x=1 x=-1$$