Solve sqrt{x}*2+3x=9

$$\sqrt { x } \cdot 2 + 3 x = 9$$

Solve for x

$$x = \frac{29 - 4 \sqrt{7}}{9} \approx 2.046332751$$

Solution Steps

Subtract 3x from both sides of the equation.

$$2\sqrt{x}=9-3x$$

Square both sides of the equation.

$$\left(2\sqrt{x}\right)^{2}=\left(9-3x\right)^{2}$$

Expand \left(2\sqrt{x}\right)^{2}.

$$2^{2}\left(\sqrt{x}\right)^{2}=\left(9-3x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x}\right)^{2}=\left(9-3x\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$4x=\left(9-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9-3x\right)^{2}.

$$4x=81-54x+9x^{2}$$

Subtract 81 from both sides.

$$4x-81=-54x+9x^{2}$$

Add 54x to both sides.

$$4x-81+54x=9x^{2}$$

Combine 4x and 54x to get 58x.

$$58x-81=9x^{2}$$

Subtract 9x^{2} from both sides.

$$58x-81-9x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-9x^{2}+58x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 58 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-58±\sqrt{58^{2}-4\left(-9\right)\left(-81\right)}}{2\left(-9\right)}$$

Square 58.

$$x=\frac{-58±\sqrt{3364-4\left(-9\right)\left(-81\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-58±\sqrt{3364+36\left(-81\right)}}{2\left(-9\right)}$$

Multiply 36 times -81.

$$x=\frac{-58±\sqrt{3364-2916}}{2\left(-9\right)}$$

Add 3364 to -2916.

$$x=\frac{-58±\sqrt{448}}{2\left(-9\right)}$$

Take the square root of 448.

$$x=\frac{-58±8\sqrt{7}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-58±8\sqrt{7}}{-18}$$

Now solve the equation x=\frac{-58±8\sqrt{7}}{-18} when ± is plus. Add -58 to 8\sqrt{7}.

$$x=\frac{8\sqrt{7}-58}{-18}$$

Divide -58+8\sqrt{7} by -18.

$$x=\frac{29-4\sqrt{7}}{9}$$

Now solve the equation x=\frac{-58±8\sqrt{7}}{-18} when ± is minus. Subtract 8\sqrt{7} from -58.

$$x=\frac{-8\sqrt{7}-58}{-18}$$

Divide -58-8\sqrt{7} by -18.

$$x=\frac{4\sqrt{7}+29}{9}$$

The equation is now solved.

$$x=\frac{29-4\sqrt{7}}{9} x=\frac{4\sqrt{7}+29}{9}$$

Substitute \frac{29-4\sqrt{7}}{9} for x in the equation 2\sqrt{x}+3x=9.

$$2\sqrt{\frac{29-4\sqrt{7}}{9}}+3\times \frac{29-4\sqrt{7}}{9}=9$$

Simplify. The value x=\frac{29-4\sqrt{7}}{9} satisfies the equation.

$$9=9$$

Substitute \frac{4\sqrt{7}+29}{9} for x in the equation 2\sqrt{x}+3x=9.

$$2\sqrt{\frac{4\sqrt{7}+29}{9}}+3\times \frac{4\sqrt{7}+29}{9}=9$$

Simplify. The value x=\frac{4\sqrt{7}+29}{9} does not satisfy the equation.

$$\frac{8}{3}\times 7^{\frac{1}{2}}+\frac{31}{3}=9$$

Equation 2\sqrt{x}=9-3x has a unique solution.

$$x=\frac{29-4\sqrt{7}}{9}$$