Solve +5.4x+sqrt{x}=7.5

$$+5.4x+ \sqrt{ x } = 7.5$$

Solve for x

$$x=-\frac{25\sqrt{163}}{1458}+\frac{1025}{729}\approx 1.187120279$$

Solution Steps

Subtract 5.4x from both sides of the equation.

$$\sqrt{x}=7.5-5.4x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(7.5-5.4x\right)^{2}$$

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

$$x=\left(7.5-5.4x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7.5-5.4x\right)^{2}.

$$x=56.25-81x+29.16x^{2}$$

Subtract 56.25 from both sides.

$$x-56.25=-81x+29.16x^{2}$$

Add 81x to both sides.

$$x-56.25+81x=29.16x^{2}$$

Combine x and 81x to get 82x.

$$82x-56.25=29.16x^{2}$$

Subtract 29.16x^{2} from both sides.

$$82x-56.25-29.16x^{2}=0$$

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.

$$-29.16x^{2}+82x-56.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -29.16 for a, 82 for b, and -56.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-82±\sqrt{82^{2}-4\left(-29.16\right)\left(-56.25\right)}}{2\left(-29.16\right)}$$

Square 82.

$$x=\frac{-82±\sqrt{6724-4\left(-29.16\right)\left(-56.25\right)}}{2\left(-29.16\right)}$$

Multiply -4 times -29.16.

$$x=\frac{-82±\sqrt{6724+116.64\left(-56.25\right)}}{2\left(-29.16\right)}$$

Multiply 116.64 times -56.25 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-82±\sqrt{6724-6561}}{2\left(-29.16\right)}$$

Add 6724 to -6561.

$$x=\frac{-82±\sqrt{163}}{2\left(-29.16\right)}$$

Multiply 2 times -29.16.

$$x=\frac{-82±\sqrt{163}}{-58.32}$$

Now solve the equation x=\frac{-82±\sqrt{163}}{-58.32} when ± is plus. Add -82 to \sqrt{163}.

$$x=\frac{\sqrt{163}-82}{-58.32}$$

Divide -82+\sqrt{163} by -58.32 by multiplying -82+\sqrt{163} by the reciprocal of -58.32.

$$x=-\frac{25\sqrt{163}}{1458}+\frac{1025}{729}$$

Now solve the equation x=\frac{-82±\sqrt{163}}{-58.32} when ± is minus. Subtract \sqrt{163} from -82.

$$x=\frac{-\sqrt{163}-82}{-58.32}$$

Divide -82-\sqrt{163} by -58.32 by multiplying -82-\sqrt{163} by the reciprocal of -58.32.

$$x=\frac{25\sqrt{163}}{1458}+\frac{1025}{729}$$

The equation is now solved.

$$x=-\frac{25\sqrt{163}}{1458}+\frac{1025}{729} x=\frac{25\sqrt{163}}{1458}+\frac{1025}{729}$$

Substitute -\frac{25\sqrt{163}}{1458}+\frac{1025}{729} for x in the equation 5.4x+\sqrt{x}=7.5.

$$5.4\left(-\frac{25\sqrt{163}}{1458}+\frac{1025}{729}\right)+\sqrt{-\frac{25\sqrt{163}}{1458}+\frac{1025}{729}}=7.5$$

Simplify. The value x=-\frac{25\sqrt{163}}{1458}+\frac{1025}{729} satisfies the equation.

$$\frac{15}{2}=7.5$$

Substitute \frac{25\sqrt{163}}{1458}+\frac{1025}{729} for x in the equation 5.4x+\sqrt{x}=7.5.

$$5.4\left(\frac{25\sqrt{163}}{1458}+\frac{1025}{729}\right)+\sqrt{\frac{25\sqrt{163}}{1458}+\frac{1025}{729}}=7.5$$

Simplify. The value x=\frac{25\sqrt{163}}{1458}+\frac{1025}{729} does not satisfy the equation.

$$\frac{5}{27}\times 163^{\frac{1}{2}}+\frac{415}{54}=7.5$$

Equation \sqrt{x}=-\frac{27x}{5}+7.5 has a unique solution.

$$x=-\frac{25\sqrt{163}}{1458}+\frac{1025}{729}$$