Solve 3sqrt{x}+x=7

$$3 \sqrt { x } + x = 7$$

Solve for x

$$x = \frac{23 - 3 \sqrt{37}}{2} \approx 2.375856205$$

Solution Steps

Subtract x from both sides of the equation.

$$3\sqrt{x}=7-x$$

Square both sides of the equation.

$$\left(3\sqrt{x}\right)^{2}=\left(7-x\right)^{2}$$

Expand \left(3\sqrt{x}\right)^{2}.

$$3^{2}\left(\sqrt{x}\right)^{2}=\left(7-x\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x}\right)^{2}=\left(7-x\right)^{2}$$

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

$$9x=\left(7-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-x\right)^{2}.

$$9x=49-14x+x^{2}$$

Subtract 49 from both sides.

$$9x-49=-14x+x^{2}$$

Add 14x to both sides.

$$9x-49+14x=x^{2}$$

Combine 9x and 14x to get 23x.

$$23x-49=x^{2}$$

Subtract x^{2} from both sides.

$$23x-49-x^{2}=0$$

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.

$$-x^{2}+23x-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 23 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-23±\sqrt{529+4\left(-49\right)}}{2\left(-1\right)}$$

Multiply 4 times -49.

$$x=\frac{-23±\sqrt{529-196}}{2\left(-1\right)}$$

Add 529 to -196.

$$x=\frac{-23±\sqrt{333}}{2\left(-1\right)}$$

Take the square root of 333.

$$x=\frac{-23±3\sqrt{37}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-23±3\sqrt{37}}{-2}$$

Now solve the equation x=\frac{-23±3\sqrt{37}}{-2} when ± is plus. Add -23 to 3\sqrt{37}.

$$x=\frac{3\sqrt{37}-23}{-2}$$

Divide -23+3\sqrt{37} by -2.

$$x=\frac{23-3\sqrt{37}}{2}$$

Now solve the equation x=\frac{-23±3\sqrt{37}}{-2} when ± is minus. Subtract 3\sqrt{37} from -23.

$$x=\frac{-3\sqrt{37}-23}{-2}$$

Divide -23-3\sqrt{37} by -2.

$$x=\frac{3\sqrt{37}+23}{2}$$

The equation is now solved.

$$x=\frac{23-3\sqrt{37}}{2} x=\frac{3\sqrt{37}+23}{2}$$

Substitute \frac{23-3\sqrt{37}}{2} for x in the equation 3\sqrt{x}+x=7.

$$3\sqrt{\frac{23-3\sqrt{37}}{2}}+\frac{23-3\sqrt{37}}{2}=7$$

Simplify. The value x=\frac{23-3\sqrt{37}}{2} satisfies the equation.

$$7=7$$

Substitute \frac{3\sqrt{37}+23}{2} for x in the equation 3\sqrt{x}+x=7.

$$3\sqrt{\frac{3\sqrt{37}+23}{2}}+\frac{3\sqrt{37}+23}{2}=7$$

Simplify. The value x=\frac{3\sqrt{37}+23}{2} does not satisfy the equation.

$$16+3\times 37^{\frac{1}{2}}=7$$

Equation 3\sqrt{x}=7-x has a unique solution.

$$x=\frac{23-3\sqrt{37}}{2}$$