Solve x^2+(3x/4+1)^2-2(3x/4+1)=24

$$x ^ { 2 } + ( \frac { 3 x } { 4 } + 1 ) ^ { 2 } - 2 ( \frac { 3 x } { 4 } + 1 ) = 24$$

Solve for x

$$x=-4$$

$$x=4$$

Steps Using Difference of Squares

Multiply both sides of the equation by 4.

$$4\left(x^{2}+\left(\frac{3x}{4}+1\right)^{2}\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3x}{4}+1\right)^{2}.

$$4\left(x^{2}+\left(\frac{3x}{4}\right)^{2}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To raise \frac{3x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 4, the greatest common factor in 2 and 4.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+\frac{3x}{2}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 2 is 16. Multiply \frac{3x}{2} times \frac{8}{8}.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{16}+\frac{8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Since \frac{\left(3x\right)^{2}}{16} and \frac{8\times 3x}{16} have the same denominator, add them by adding their numerators.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Do the multiplications in \left(3x\right)^{2}+8\times 3x.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Combine like terms in \left(3x\right)^{2}+24x.

$$4\left(x^{2}+\frac{9x^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use the distributive property to multiply 4 by x^{2}+\frac{9x^{2}+24x}{16}+1.

$$4x^{2}+4\times \frac{9x^{2}+24x}{16}+4-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 16, the greatest common factor in 4 and 16.

$$4x^{2}+\frac{9x^{2}+24x}{4}+4-8\left(\frac{3x}{4}+1\right)=96$$

Divide each term of 9x^{2}+24x by 4 to get \frac{9}{4}x^{2}+6x.

$$4x^{2}+\frac{9}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Combine 4x^{2} and \frac{9}{4}x^{2} to get \frac{25}{4}x^{2}.

$$\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Subtract 96 from both sides.

$$\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)-96=0$$

Multiply both sides of the equation by 4.

$$4\left(\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)\right)-384=0$$

Multiply both sides of the equation by 4.

$$16\left(\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)\right)-1536=0$$

Use the distributive property to multiply -8 by \frac{3x}{4}+1.

$$16\left(\frac{25}{4}x^{2}+6x+4-8\times \frac{3x}{4}-8\right)-1536=0$$

Cancel out 4, the greatest common factor in 8 and 4.

$$16\left(\frac{25}{4}x^{2}+6x+4-2\times 3x-8\right)-1536=0$$

Multiply -2 and 3 to get -6.

$$16\left(\frac{25}{4}x^{2}+6x+4-6x-8\right)-1536=0$$

Combine 6x and -6x to get 0.

$$16\left(\frac{25}{4}x^{2}+4-8\right)-1536=0$$

Subtract 8 from 4 to get -4.

$$16\left(\frac{25}{4}x^{2}-4\right)-1536=0$$

Use the distributive property to multiply 16 by \frac{25}{4}x^{2}-4.

$$100x^{2}-64-1536=0$$

Subtract 1536 from -64 to get -1600.

$$100x^{2}-1600=0$$

Divide both sides by 100.

$$x^{2}-16=0$$

Consider x^{2}-16. Rewrite x^{2}-16 as x^{2}-4^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-4\right)\left(x+4\right)=0$$

To find equation solutions, solve x-4=0 and x+4=0.

$$x=4 x=-4$$

Steps by Finding Square Root

Multiply both sides of the equation by 4.

$$4\left(x^{2}+\left(\frac{3x}{4}+1\right)^{2}\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3x}{4}+1\right)^{2}.

$$4\left(x^{2}+\left(\frac{3x}{4}\right)^{2}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To raise \frac{3x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 4, the greatest common factor in 2 and 4.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+\frac{3x}{2}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 2 is 16. Multiply \frac{3x}{2} times \frac{8}{8}.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{16}+\frac{8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Since \frac{\left(3x\right)^{2}}{16} and \frac{8\times 3x}{16} have the same denominator, add them by adding their numerators.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Do the multiplications in \left(3x\right)^{2}+8\times 3x.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Combine like terms in \left(3x\right)^{2}+24x.

$$4\left(x^{2}+\frac{9x^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use the distributive property to multiply 4 by x^{2}+\frac{9x^{2}+24x}{16}+1.

$$4x^{2}+4\times \frac{9x^{2}+24x}{16}+4-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 16, the greatest common factor in 4 and 16.

$$4x^{2}+\frac{9x^{2}+24x}{4}+4-8\left(\frac{3x}{4}+1\right)=96$$

Divide each term of 9x^{2}+24x by 4 to get \frac{9}{4}x^{2}+6x.

$$4x^{2}+\frac{9}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Combine 4x^{2} and \frac{9}{4}x^{2} to get \frac{25}{4}x^{2}.

$$\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Multiply both sides of the equation by 4.

$$4\left(\frac{25}{4}x^{2}+6x+4\right)-32\left(\frac{3x}{4}+1\right)=384$$

Multiply both sides of the equation by 4.

$$16\left(\frac{25}{4}x^{2}+6x+4\right)-4\times 32\left(\frac{3x}{4}+1\right)=1536$$

Use the distributive property to multiply 16 by \frac{25}{4}x^{2}+6x+4.

$$100x^{2}+96x+64-4\times 32\left(\frac{3x}{4}+1\right)=1536$$

Multiply -4 and 32 to get -128.

$$100x^{2}+96x+64-128\left(\frac{3x}{4}+1\right)=1536$$

Use the distributive property to multiply -128 by \frac{3x}{4}+1.

$$100x^{2}+96x+64-128\times \frac{3x}{4}-128=1536$$

Cancel out 4, the greatest common factor in 128 and 4.

$$100x^{2}+96x+64-32\times 3x-128=1536$$

Multiply -32 and 3 to get -96.

$$100x^{2}+96x+64-96x-128=1536$$

Combine 96x and -96x to get 0.

$$100x^{2}+64-128=1536$$

Subtract 128 from 64 to get -64.

$$100x^{2}-64=1536$$

Add 64 to both sides.

$$100x^{2}=1536+64$$

Add 1536 and 64 to get 1600.

$$100x^{2}=1600$$

Divide both sides by 100.

$$x^{2}=\frac{1600}{100}$$

Divide 1600 by 100 to get 16.

$$x^{2}=16$$

Take the square root of both sides of the equation.

$$x=4 x=-4$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$4\left(x^{2}+\left(\frac{3x}{4}+1\right)^{2}\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3x}{4}+1\right)^{2}.

$$4\left(x^{2}+\left(\frac{3x}{4}\right)^{2}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To raise \frac{3x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+2\times \frac{3x}{4}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 4, the greatest common factor in 2 and 4.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{4^{2}}+\frac{3x}{2}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 2 is 16. Multiply \frac{3x}{2} times \frac{8}{8}.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}}{16}+\frac{8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Since \frac{\left(3x\right)^{2}}{16} and \frac{8\times 3x}{16} have the same denominator, add them by adding their numerators.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+8\times 3x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Do the multiplications in \left(3x\right)^{2}+8\times 3x.

$$4\left(x^{2}+\frac{\left(3x\right)^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Combine like terms in \left(3x\right)^{2}+24x.

$$4\left(x^{2}+\frac{9x^{2}+24x}{16}+1\right)-8\left(\frac{3x}{4}+1\right)=96$$

Use the distributive property to multiply 4 by x^{2}+\frac{9x^{2}+24x}{16}+1.

$$4x^{2}+4\times \frac{9x^{2}+24x}{16}+4-8\left(\frac{3x}{4}+1\right)=96$$

Cancel out 16, the greatest common factor in 4 and 16.

$$4x^{2}+\frac{9x^{2}+24x}{4}+4-8\left(\frac{3x}{4}+1\right)=96$$

Divide each term of 9x^{2}+24x by 4 to get \frac{9}{4}x^{2}+6x.

$$4x^{2}+\frac{9}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Combine 4x^{2} and \frac{9}{4}x^{2} to get \frac{25}{4}x^{2}.

$$\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)=96$$

Subtract 96 from both sides.

$$\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)-96=0$$

Multiply both sides of the equation by 4.

$$4\left(\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)\right)-384=0$$

Multiply both sides of the equation by 4.

$$16\left(\frac{25}{4}x^{2}+6x+4-8\left(\frac{3x}{4}+1\right)\right)-1536=0$$

Use the distributive property to multiply -8 by \frac{3x}{4}+1.

$$16\left(\frac{25}{4}x^{2}+6x+4-8\times \frac{3x}{4}-8\right)-1536=0$$

Cancel out 4, the greatest common factor in 8 and 4.

$$16\left(\frac{25}{4}x^{2}+6x+4-2\times 3x-8\right)-1536=0$$

Multiply -2 and 3 to get -6.

$$16\left(\frac{25}{4}x^{2}+6x+4-6x-8\right)-1536=0$$

Combine 6x and -6x to get 0.

$$16\left(\frac{25}{4}x^{2}+4-8\right)-1536=0$$

Subtract 8 from 4 to get -4.

$$16\left(\frac{25}{4}x^{2}-4\right)-1536=0$$

Use the distributive property to multiply 16 by \frac{25}{4}x^{2}-4.

$$100x^{2}-64-1536=0$$

Subtract 1536 from -64 to get -1600.

$$100x^{2}-1600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 0 for b, and -1600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 100\left(-1600\right)}}{2\times 100}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 100\left(-1600\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{0±\sqrt{-400\left(-1600\right)}}{2\times 100}$$

Multiply -400 times -1600.

$$x=\frac{0±\sqrt{640000}}{2\times 100}$$

Take the square root of 640000.

$$x=\frac{0±800}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{0±800}{200}$$

Now solve the equation x=\frac{0±800}{200} when ± is plus. Divide 800 by 200.

$$x=4$$

Now solve the equation x=\frac{0±800}{200} when ± is minus. Divide -800 by 200.

$$x=-4$$

The equation is now solved.

$$x=4 x=-4$$