Solve 3666x+sqrt{4x}=59

$$3666x+ \sqrt{ 4x } =59$$

Solve for x

$$x=-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}\approx 0.016024774$$

Solution Steps

Subtract 3666x from both sides of the equation.

$$\sqrt{4x}=59-3666x$$

Square both sides of the equation.

$$\left(\sqrt{4x}\right)^{2}=\left(59-3666x\right)^{2}$$

Calculate \sqrt{4x} to the power of 2 and get 4x.

$$4x=\left(59-3666x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(59-3666x\right)^{2}.

$$4x=3481-432588x+13439556x^{2}$$

Subtract 3481 from both sides.

$$4x-3481=-432588x+13439556x^{2}$$

Add 432588x to both sides.

$$4x-3481+432588x=13439556x^{2}$$

Combine 4x and 432588x to get 432592x.

$$432592x-3481=13439556x^{2}$$

Subtract 13439556x^{2} from both sides.

$$432592x-3481-13439556x^{2}=0$$

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.

$$-13439556x^{2}+432592x-3481=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13439556 for a, 432592 for b, and -3481 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-432592±\sqrt{432592^{2}-4\left(-13439556\right)\left(-3481\right)}}{2\left(-13439556\right)}$$

Square 432592.

$$x=\frac{-432592±\sqrt{187135838464-4\left(-13439556\right)\left(-3481\right)}}{2\left(-13439556\right)}$$

Multiply -4 times -13439556.

$$x=\frac{-432592±\sqrt{187135838464+53758224\left(-3481\right)}}{2\left(-13439556\right)}$$

Multiply 53758224 times -3481.

$$x=\frac{-432592±\sqrt{187135838464-187132377744}}{2\left(-13439556\right)}$$

Add 187135838464 to -187132377744.

$$x=\frac{-432592±\sqrt{3460720}}{2\left(-13439556\right)}$$

Take the square root of 3460720.

$$x=\frac{-432592±4\sqrt{216295}}{2\left(-13439556\right)}$$

Multiply 2 times -13439556.

$$x=\frac{-432592±4\sqrt{216295}}{-26879112}$$

Now solve the equation x=\frac{-432592±4\sqrt{216295}}{-26879112} when ± is plus. Add -432592 to 4\sqrt{216295}.

$$x=\frac{4\sqrt{216295}-432592}{-26879112}$$

Divide -432592+4\sqrt{216295} by -26879112.

$$x=-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}$$

Now solve the equation x=\frac{-432592±4\sqrt{216295}}{-26879112} when ± is minus. Subtract 4\sqrt{216295} from -432592.

$$x=\frac{-4\sqrt{216295}-432592}{-26879112}$$

Divide -432592-4\sqrt{216295} by -26879112.

$$x=\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}$$

The equation is now solved.

$$x=-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889} x=\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}$$

Substitute -\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889} for x in the equation 3666x+\sqrt{4x}=59.

$$3666\left(-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}\right)+\sqrt{4\left(-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}\right)}=59$$

Simplify. The value x=-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889} satisfies the equation.

$$59=59$$

Substitute \frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889} for x in the equation 3666x+\sqrt{4x}=59.

$$3666\left(\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}\right)+\sqrt{4\left(\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}\right)}=59$$

Simplify. The value x=\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889} does not satisfy the equation.

$$\frac{2}{1833}\times 216295^{\frac{1}{2}}+\frac{108149}{1833}=59$$

Equation \sqrt{4x}=59-3666x has a unique solution.

$$x=-\frac{\sqrt{216295}}{6719778}+\frac{54074}{3359889}$$