Solve (x+1/2)^2=1/3*3x/2+5/4

$$( \frac { x + 1 } { 2 } ) ^ { 2 } = \frac { 1 } { 3 } \cdot \frac { 3 x } { 2 } + \frac { 5 } { 4 }$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using Difference of Squares

Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$12\times \left(\frac{x+1}{2}\right)^{2}=2\times 3x+15$$

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

$$12\times \frac{\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Express 12\times \frac{\left(x+1\right)^{2}}{2^{2}} as a single fraction.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Multiply 2 and 3 to get 6.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=6x+15$$

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

$$\frac{12\left(x^{2}+2x+1\right)}{2^{2}}=6x+15$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12\left(x^{2}+2x+1\right)}{4}=6x+15$$

Divide 12\left(x^{2}+2x+1\right) by 4 to get 3\left(x^{2}+2x+1\right).

$$3\left(x^{2}+2x+1\right)=6x+15$$

Use the distributive property to multiply 3 by x^{2}+2x+1.

$$3x^{2}+6x+3=6x+15$$

Subtract 6x from both sides.

$$3x^{2}+6x+3-6x=15$$

Combine 6x and -6x to get 0.

$$3x^{2}+3=15$$

Subtract 15 from both sides.

$$3x^{2}+3-15=0$$

Subtract 15 from 3 to get -12.

$$3x^{2}-12=0$$

Divide both sides by 3.

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

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

$$\left(x-2\right)\left(x+2\right)=0$$

To find equation solutions, solve x-2=0 and x+2=0.

$$x=2 x=-2$$

Steps by Finding Square Root

Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$12\times \left(\frac{x+1}{2}\right)^{2}=2\times 3x+15$$

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

$$12\times \frac{\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Express 12\times \frac{\left(x+1\right)^{2}}{2^{2}} as a single fraction.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Multiply 2 and 3 to get 6.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=6x+15$$

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

$$\frac{12\left(x^{2}+2x+1\right)}{2^{2}}=6x+15$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12\left(x^{2}+2x+1\right)}{4}=6x+15$$

Divide 12\left(x^{2}+2x+1\right) by 4 to get 3\left(x^{2}+2x+1\right).

$$3\left(x^{2}+2x+1\right)=6x+15$$

Use the distributive property to multiply 3 by x^{2}+2x+1.

$$3x^{2}+6x+3=6x+15$$

Subtract 6x from both sides.

$$3x^{2}+6x+3-6x=15$$

Combine 6x and -6x to get 0.

$$3x^{2}+3=15$$

Subtract 3 from both sides.

$$3x^{2}=15-3$$

Subtract 3 from 15 to get 12.

$$3x^{2}=12$$

Divide both sides by 3.

$$x^{2}=\frac{12}{3}$$

Divide 12 by 3 to get 4.

$$x^{2}=4$$

Take the square root of both sides of the equation.

$$x=2 x=-2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$12\times \left(\frac{x+1}{2}\right)^{2}=2\times 3x+15$$

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

$$12\times \frac{\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Express 12\times \frac{\left(x+1\right)^{2}}{2^{2}} as a single fraction.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=2\times 3x+15$$

Multiply 2 and 3 to get 6.

$$\frac{12\left(x+1\right)^{2}}{2^{2}}=6x+15$$

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

$$\frac{12\left(x^{2}+2x+1\right)}{2^{2}}=6x+15$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12\left(x^{2}+2x+1\right)}{4}=6x+15$$

Divide 12\left(x^{2}+2x+1\right) by 4 to get 3\left(x^{2}+2x+1\right).

$$3\left(x^{2}+2x+1\right)=6x+15$$

Use the distributive property to multiply 3 by x^{2}+2x+1.

$$3x^{2}+6x+3=6x+15$$

Subtract 6x from both sides.

$$3x^{2}+6x+3-6x=15$$

Combine 6x and -6x to get 0.

$$3x^{2}+3=15$$

Subtract 15 from both sides.

$$3x^{2}+3-15=0$$

Subtract 15 from 3 to get -12.

$$3x^{2}-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{0±\sqrt{144}}{2\times 3}$$

Take the square root of 144.

$$x=\frac{0±12}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±12}{6}$$

Now solve the equation x=\frac{0±12}{6} when ± is plus. Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{0±12}{6} when ± is minus. Divide -12 by 6.

$$x=-2$$

The equation is now solved.

$$x=2 x=-2$$