Solve x=sqrt{76.5-4*5x}

$$x = \sqrt{ 76.5-4 \cdot 5x }$$

Solve for x

$$x = \frac{\sqrt{706} - 20}{2} \approx 3.285330256$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{76.5-4\times 5x}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$x^{2}=\left(\sqrt{76.5-20x}\right)^{2}$$

Calculate \sqrt{76.5-20x} to the power of 2 and get 76.5-20x.

$$x^{2}=76.5-20x$$

Subtract 76.5 from both sides.

$$x^{2}-76.5=-20x$$

Add 20x to both sides.

$$x^{2}-76.5+20x=0$$

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.

$$x^{2}+20x-76.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -76.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-76.5\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-76.5\right)}}{2}$$

Multiply -4 times -76.5.

$$x=\frac{-20±\sqrt{400+306}}{2}$$

Add 400 to 306.

$$x=\frac{-20±\sqrt{706}}{2}$$

Now solve the equation x=\frac{-20±\sqrt{706}}{2} when ± is plus. Add -20 to \sqrt{706}.

$$x=\frac{\sqrt{706}-20}{2}$$

Divide -20+\sqrt{706} by 2.

$$x=\frac{\sqrt{706}}{2}-10$$

Now solve the equation x=\frac{-20±\sqrt{706}}{2} when ± is minus. Subtract \sqrt{706} from -20.

$$x=\frac{-\sqrt{706}-20}{2}$$

Divide -20-\sqrt{706} by 2.

$$x=-\frac{\sqrt{706}}{2}-10$$

The equation is now solved.

$$x=\frac{\sqrt{706}}{2}-10 x=-\frac{\sqrt{706}}{2}-10$$

Substitute \frac{\sqrt{706}}{2}-10 for x in the equation x=\sqrt{76.5-4\times 5x}.

$$\frac{\sqrt{706}}{2}-10=\sqrt{76.5-4\times 5\left(\frac{\sqrt{706}}{2}-10\right)}$$

Simplify. The value x=\frac{\sqrt{706}}{2}-10 satisfies the equation.

$$-10+\frac{1}{2}\times 706^{\frac{1}{2}}=\frac{1}{2}\times 706^{\frac{1}{2}}-10$$

Substitute -\frac{\sqrt{706}}{2}-10 for x in the equation x=\sqrt{76.5-4\times 5x}.

$$-\frac{\sqrt{706}}{2}-10=\sqrt{76.5-4\times 5\left(-\frac{\sqrt{706}}{2}-10\right)}$$

Simplify. The value x=-\frac{\sqrt{706}}{2}-10 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{1}{2}\times 706^{\frac{1}{2}}-10=\frac{1}{2}\times 706^{\frac{1}{2}}+10$$

Equation x=\sqrt{76.5-20x} has a unique solution.

$$x=\frac{\sqrt{706}}{2}-10$$