Solve 0.5(x+6)+x=sqrt{0.75x^2-6x+36}

$$0.5(x+6)+x= \sqrt{ 0.75 { x }^{ 2 } -6x+36 }$$

Solve for x

$$x=\sqrt{43}-5\approx 1.557438524$$

Solution Steps

Square both sides of the equation.

$$\left(0.5\left(x+6\right)+x\right)^{2}=\left(\sqrt{0.75x^{2}-6x+36}\right)^{2}$$

Use the distributive property to multiply 0.5 by x+6.

$$\left(0.5x+3+x\right)^{2}=\left(\sqrt{0.75x^{2}-6x+36}\right)^{2}$$

Combine 0.5x and x to get 1.5x.

$$\left(1.5x+3\right)^{2}=\left(\sqrt{0.75x^{2}-6x+36}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1.5x+3\right)^{2}.

$$2.25x^{2}+9x+9=\left(\sqrt{0.75x^{2}-6x+36}\right)^{2}$$

Calculate \sqrt{0.75x^{2}-6x+36} to the power of 2 and get 0.75x^{2}-6x+36.

$$2.25x^{2}+9x+9=0.75x^{2}-6x+36$$

Subtract 0.75x^{2} from both sides.

$$2.25x^{2}+9x+9-0.75x^{2}=-6x+36$$

Combine 2.25x^{2} and -0.75x^{2} to get 1.5x^{2}.

$$1.5x^{2}+9x+9=-6x+36$$

Add 6x to both sides.

$$1.5x^{2}+9x+9+6x=36$$

Combine 9x and 6x to get 15x.

$$1.5x^{2}+15x+9=36$$

Subtract 36 from both sides.

$$1.5x^{2}+15x+9-36=0$$

Subtract 36 from 9 to get -27.

$$1.5x^{2}+15x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5 for a, 15 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 1.5\left(-27\right)}}{2\times 1.5}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 1.5\left(-27\right)}}{2\times 1.5}$$

Multiply -4 times 1.5.

$$x=\frac{-15±\sqrt{225-6\left(-27\right)}}{2\times 1.5}$$

Multiply -6 times -27.

$$x=\frac{-15±\sqrt{225+162}}{2\times 1.5}$$

Add 225 to 162.

$$x=\frac{-15±\sqrt{387}}{2\times 1.5}$$

Take the square root of 387.

$$x=\frac{-15±3\sqrt{43}}{2\times 1.5}$$

Multiply 2 times 1.5.

$$x=\frac{-15±3\sqrt{43}}{3}$$

Now solve the equation x=\frac{-15±3\sqrt{43}}{3} when ± is plus. Add -15 to 3\sqrt{43}.

$$x=\frac{3\sqrt{43}-15}{3}$$

Divide -15+3\sqrt{43} by 3.

$$x=\sqrt{43}-5$$

Now solve the equation x=\frac{-15±3\sqrt{43}}{3} when ± is minus. Subtract 3\sqrt{43} from -15.

$$x=\frac{-3\sqrt{43}-15}{3}$$

Divide -15-3\sqrt{43} by 3.

$$x=-\sqrt{43}-5$$

The equation is now solved.

$$x=\sqrt{43}-5 x=-\sqrt{43}-5$$

Substitute \sqrt{43}-5 for x in the equation 0.5\left(x+6\right)+x=\sqrt{0.75x^{2}-6x+36}.

$$0.5\left(\sqrt{43}-5+6\right)+\sqrt{43}-5=\sqrt{0.75\left(\sqrt{43}-5\right)^{2}-6\left(\sqrt{43}-5\right)+36}$$

Simplify. The value x=\sqrt{43}-5 satisfies the equation.

$$1.5\times 43^{\frac{1}{2}}-4.5=-\left(\frac{9}{2}-\frac{3}{2}\times 43^{\frac{1}{2}}\right)$$

Substitute -\sqrt{43}-5 for x in the equation 0.5\left(x+6\right)+x=\sqrt{0.75x^{2}-6x+36}.

$$0.5\left(-\sqrt{43}-5+6\right)-\sqrt{43}-5=\sqrt{0.75\left(-\sqrt{43}-5\right)^{2}-6\left(-\sqrt{43}-5\right)+36}$$

Simplify. The value x=-\sqrt{43}-5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1.5\times 43^{\frac{1}{2}}-4.5=\frac{9}{2}+\frac{3}{2}\times 43^{\frac{1}{2}}$$

Equation \frac{x+6}{2}+x=\sqrt{\frac{3x^{2}}{4}-6x+36} has a unique solution.

$$x=\sqrt{43}-5$$