Solve 29x+4sqrt{x}=480

$$29x+4 \sqrt{ x } =480$$

Solve for x

$$x=16$$

Solution Steps

Subtract 29x from both sides of the equation.

$$4\sqrt{x}=480-29x$$

Square both sides of the equation.

$$\left(4\sqrt{x}\right)^{2}=\left(480-29x\right)^{2}$$

Expand \left(4\sqrt{x}\right)^{2}.

$$4^{2}\left(\sqrt{x}\right)^{2}=\left(480-29x\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x}\right)^{2}=\left(480-29x\right)^{2}$$

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

$$16x=\left(480-29x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(480-29x\right)^{2}.

$$16x=230400-27840x+841x^{2}$$

Subtract 230400 from both sides.

$$16x-230400=-27840x+841x^{2}$$

Add 27840x to both sides.

$$16x-230400+27840x=841x^{2}$$

Combine 16x and 27840x to get 27856x.

$$27856x-230400=841x^{2}$$

Subtract 841x^{2} from both sides.

$$27856x-230400-841x^{2}=0$$

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.

$$-841x^{2}+27856x-230400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -841 for a, 27856 for b, and -230400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27856±\sqrt{27856^{2}-4\left(-841\right)\left(-230400\right)}}{2\left(-841\right)}$$

Square 27856.

$$x=\frac{-27856±\sqrt{775956736-4\left(-841\right)\left(-230400\right)}}{2\left(-841\right)}$$

Multiply -4 times -841.

$$x=\frac{-27856±\sqrt{775956736+3364\left(-230400\right)}}{2\left(-841\right)}$$

Multiply 3364 times -230400.

$$x=\frac{-27856±\sqrt{775956736-775065600}}{2\left(-841\right)}$$

Add 775956736 to -775065600.

$$x=\frac{-27856±\sqrt{891136}}{2\left(-841\right)}$$

Take the square root of 891136.

$$x=\frac{-27856±944}{2\left(-841\right)}$$

Multiply 2 times -841.

$$x=\frac{-27856±944}{-1682}$$

Now solve the equation x=\frac{-27856±944}{-1682} when ± is plus. Add -27856 to 944.

$$x=-\frac{26912}{-1682}$$

Divide -26912 by -1682.

$$x=16$$

Now solve the equation x=\frac{-27856±944}{-1682} when ± is minus. Subtract 944 from -27856.

$$x=-\frac{28800}{-1682}$$

Reduce the fraction \frac{-28800}{-1682} to lowest terms by extracting and canceling out 2.

$$x=\frac{14400}{841}$$

The equation is now solved.

$$x=16 x=\frac{14400}{841}$$

Substitute 16 for x in the equation 29x+4\sqrt{x}=480.

$$29\times 16+4\sqrt{16}=480$$

Simplify. The value x=16 satisfies the equation.

$$480=480$$

Substitute \frac{14400}{841} for x in the equation 29x+4\sqrt{x}=480.

$$29\times \frac{14400}{841}+4\sqrt{\frac{14400}{841}}=480$$

Simplify. The value x=\frac{14400}{841} does not satisfy the equation.

$$\frac{14880}{29}=480$$

Equation 4\sqrt{x}=480-29x has a unique solution.

$$x=16$$