Solve 3x-8sqrt{x}-60=0

$$3x-8 \sqrt{ x } -60=0$$

Solve for x

$$x=36$$

Solution Steps

Add 60 to both sides. Anything plus zero gives itself.

$$3x-8\sqrt{x}=60$$

Subtract 3x from both sides of the equation.

$$-8\sqrt{x}=60-3x$$

Square both sides of the equation.

$$\left(-8\sqrt{x}\right)^{2}=\left(60-3x\right)^{2}$$

Expand \left(-8\sqrt{x}\right)^{2}.

$$\left(-8\right)^{2}\left(\sqrt{x}\right)^{2}=\left(60-3x\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$64\left(\sqrt{x}\right)^{2}=\left(60-3x\right)^{2}$$

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

$$64x=\left(60-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(60-3x\right)^{2}.

$$64x=3600-360x+9x^{2}$$

Add 360x to both sides.

$$64x+360x=3600+9x^{2}$$

Combine 64x and 360x to get 424x.

$$424x=3600+9x^{2}$$

Subtract 9x^{2} from both sides.

$$424x-9x^{2}=3600$$

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.

$$-9x^{2}+424x=3600$$

Subtract 3600 from both sides of the equation.

$$-9x^{2}+424x-3600=3600-3600$$

Subtracting 3600 from itself leaves 0.

$$-9x^{2}+424x-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 424 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-424±\sqrt{424^{2}-4\left(-9\right)\left(-3600\right)}}{2\left(-9\right)}$$

Square 424.

$$x=\frac{-424±\sqrt{179776-4\left(-9\right)\left(-3600\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-424±\sqrt{179776+36\left(-3600\right)}}{2\left(-9\right)}$$

Multiply 36 times -3600.

$$x=\frac{-424±\sqrt{179776-129600}}{2\left(-9\right)}$$

Add 179776 to -129600.

$$x=\frac{-424±\sqrt{50176}}{2\left(-9\right)}$$

Take the square root of 50176.

$$x=\frac{-424±224}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-424±224}{-18}$$

Now solve the equation x=\frac{-424±224}{-18} when ± is plus. Add -424 to 224.

$$x=-\frac{200}{-18}$$

Reduce the fraction \frac{-200}{-18} to lowest terms by extracting and canceling out 2.

$$x=\frac{100}{9}$$

Now solve the equation x=\frac{-424±224}{-18} when ± is minus. Subtract 224 from -424.

$$x=-\frac{648}{-18}$$

Divide -648 by -18.

$$x=36$$

The equation is now solved.

$$x=\frac{100}{9} x=36$$

Substitute \frac{100}{9} for x in the equation 3x-8\sqrt{x}-60=0.

$$3\times \frac{100}{9}-8\sqrt{\frac{100}{9}}-60=0$$

Simplify. The value x=\frac{100}{9} does not satisfy the equation.

$$-\frac{160}{3}=0$$

Substitute 36 for x in the equation 3x-8\sqrt{x}-60=0.

$$3\times 36-8\sqrt{36}-60=0$$

Simplify. The value x=36 satisfies the equation.

$$0=0$$

Equation -8\sqrt{x}=60-3x has a unique solution.

$$x=36$$