Solve 28x=sqrt{x+5/4}

$$28x = \sqrt{ x+ \frac{ 5 }{ 4 } }$$

Solve for x

$$x=\frac{\sqrt{3921}+1}{1568}\approx 0.040572633$$

Solution Steps

Square both sides of the equation.

$$\left(28x\right)^{2}=\left(\sqrt{x+\frac{5}{4}}\right)^{2}$$

Expand \left(28x\right)^{2}.

$$28^{2}x^{2}=\left(\sqrt{x+\frac{5}{4}}\right)^{2}$$

Calculate 28 to the power of 2 and get 784.

$$784x^{2}=\left(\sqrt{x+\frac{5}{4}}\right)^{2}$$

Calculate \sqrt{x+\frac{5}{4}} to the power of 2 and get x+\frac{5}{4}.

$$784x^{2}=x+\frac{5}{4}$$

Subtract x from both sides.

$$784x^{2}-x=\frac{5}{4}$$

Subtract \frac{5}{4} from both sides.

$$784x^{2}-x-\frac{5}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 784 for a, -1 for b, and -\frac{5}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 784\left(-\frac{5}{4}\right)}}{2\times 784}$$

Multiply -4 times 784.

$$x=\frac{-\left(-1\right)±\sqrt{1-3136\left(-\frac{5}{4}\right)}}{2\times 784}$$

Multiply -3136 times -\frac{5}{4}.

$$x=\frac{-\left(-1\right)±\sqrt{1+3920}}{2\times 784}$$

Add 1 to 3920.

$$x=\frac{-\left(-1\right)±\sqrt{3921}}{2\times 784}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{3921}}{2\times 784}$$

Multiply 2 times 784.

$$x=\frac{1±\sqrt{3921}}{1568}$$

Now solve the equation x=\frac{1±\sqrt{3921}}{1568} when ± is plus. Add 1 to \sqrt{3921}.

$$x=\frac{\sqrt{3921}+1}{1568}$$

Now solve the equation x=\frac{1±\sqrt{3921}}{1568} when ± is minus. Subtract \sqrt{3921} from 1.

$$x=\frac{1-\sqrt{3921}}{1568}$$

The equation is now solved.

$$x=\frac{\sqrt{3921}+1}{1568} x=\frac{1-\sqrt{3921}}{1568}$$

Substitute \frac{\sqrt{3921}+1}{1568} for x in the equation 28x=\sqrt{x+\frac{5}{4}}.

$$28\times \frac{\sqrt{3921}+1}{1568}=\sqrt{\frac{\sqrt{3921}+1}{1568}+\frac{5}{4}}$$

Simplify. The value x=\frac{\sqrt{3921}+1}{1568} satisfies the equation.

$$\frac{1}{56}\times 3921^{\frac{1}{2}}+\frac{1}{56}=\frac{1}{56}+\frac{1}{56}\times 3921^{\frac{1}{2}}$$

Substitute \frac{1-\sqrt{3921}}{1568} for x in the equation 28x=\sqrt{x+\frac{5}{4}}.

$$28\times \frac{1-\sqrt{3921}}{1568}=\sqrt{\frac{1-\sqrt{3921}}{1568}+\frac{5}{4}}$$

Simplify. The value x=\frac{1-\sqrt{3921}}{1568} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{56}-\frac{1}{56}\times 3921^{\frac{1}{2}}=-\frac{1}{56}+\frac{1}{56}\times 3921^{\frac{1}{2}}$$

Equation 28x=\sqrt{x+\frac{5}{4}} has a unique solution.

$$x=\frac{\sqrt{3921}+1}{1568}$$