Solve 5x+.5*5*sqrt{(x^2-25)}=95

$$5x+.5 \times 5 \times \sqrt{ ( { x }^{ 2 } -25) } =95$$

Solve for x

$$x=13$$

Solution Steps

Subtract 5x from both sides of the equation.

$$0.5\times 5\sqrt{x^{2}-25}=95-5x$$

Multiply 0.5 and 5 to get 2.5.

$$2.5\sqrt{x^{2}-25}=95-5x$$

Square both sides of the equation.

$$\left(2.5\sqrt{x^{2}-25}\right)^{2}=\left(95-5x\right)^{2}$$

Expand \left(2.5\sqrt{x^{2}-25}\right)^{2}.

$$2.5^{2}\left(\sqrt{x^{2}-25}\right)^{2}=\left(95-5x\right)^{2}$$

Calculate 2.5 to the power of 2 and get 6.25.

$$6.25\left(\sqrt{x^{2}-25}\right)^{2}=\left(95-5x\right)^{2}$$

Calculate \sqrt{x^{2}-25} to the power of 2 and get x^{2}-25.

$$6.25\left(x^{2}-25\right)=\left(95-5x\right)^{2}$$

Use the distributive property to multiply 6.25 by x^{2}-25.

$$6.25x^{2}-156.25=\left(95-5x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(95-5x\right)^{2}.

$$6.25x^{2}-156.25=9025-950x+25x^{2}$$

Subtract 9025 from both sides.

$$6.25x^{2}-156.25-9025=-950x+25x^{2}$$

Subtract 9025 from -156.25 to get -9181.25.

$$6.25x^{2}-9181.25=-950x+25x^{2}$$

Add 950x to both sides.

$$6.25x^{2}-9181.25+950x=25x^{2}$$

Subtract 25x^{2} from both sides.

$$6.25x^{2}-9181.25+950x-25x^{2}=0$$

Combine 6.25x^{2} and -25x^{2} to get -18.75x^{2}.

$$-18.75x^{2}-9181.25+950x=0$$

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.

$$-18.75x^{2}+950x-9181.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18.75 for a, 950 for b, and -9181.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-950±\sqrt{950^{2}-4\left(-18.75\right)\left(-9181.25\right)}}{2\left(-18.75\right)}$$

Square 950.

$$x=\frac{-950±\sqrt{902500-4\left(-18.75\right)\left(-9181.25\right)}}{2\left(-18.75\right)}$$

Multiply -4 times -18.75.

$$x=\frac{-950±\sqrt{902500+75\left(-9181.25\right)}}{2\left(-18.75\right)}$$

Multiply 75 times -9181.25.

$$x=\frac{-950±\sqrt{902500-688593.75}}{2\left(-18.75\right)}$$

Add 902500 to -688593.75.

$$x=\frac{-950±\sqrt{213906.25}}{2\left(-18.75\right)}$$

Take the square root of 213906.25.

$$x=\frac{-950±\frac{925}{2}}{2\left(-18.75\right)}$$

Multiply 2 times -18.75.

$$x=\frac{-950±\frac{925}{2}}{-37.5}$$

Now solve the equation x=\frac{-950±\frac{925}{2}}{-37.5} when ± is plus. Add -950 to \frac{925}{2}.

$$x=-\frac{\frac{975}{2}}{-37.5}$$

Divide -\frac{975}{2} by -37.5 by multiplying -\frac{975}{2} by the reciprocal of -37.5.

$$x=13$$

Now solve the equation x=\frac{-950±\frac{925}{2}}{-37.5} when ± is minus. Subtract \frac{925}{2} from -950.

$$x=-\frac{\frac{2825}{2}}{-37.5}$$

Divide -\frac{2825}{2} by -37.5 by multiplying -\frac{2825}{2} by the reciprocal of -37.5.

$$x=\frac{113}{3}$$

The equation is now solved.

$$x=13 x=\frac{113}{3}$$

Substitute 13 for x in the equation 5x+0.5\times 5\sqrt{x^{2}-25}=95.

$$5\times 13+0.5\times 5\sqrt{13^{2}-25}=95$$

Simplify. The value x=13 satisfies the equation.

$$95=95$$

Substitute \frac{113}{3} for x in the equation 5x+0.5\times 5\sqrt{x^{2}-25}=95.

$$5\times \frac{113}{3}+0.5\times 5\sqrt{\left(\frac{113}{3}\right)^{2}-25}=95$$

Simplify. The value x=\frac{113}{3} does not satisfy the equation.

$$\frac{845}{3}=95$$

Equation \frac{5\sqrt{x^{2}-25}}{2}=95-5x has a unique solution.

$$x=13$$