Solve x^2-4/8-2x-3/5=1

$$\frac{ { x }^{ 2 } -4 }{ 8 } - \frac{ 2x-3 }{ 5 } = 1$$

Solve for x

$$x = \frac{2 \sqrt{61} + 8}{5} \approx 4.72409987$$

$$x=\frac{8-2\sqrt{61}}{5}\approx -1.52409987$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 40, the least common multiple of 8,5.

$$5\left(x^{2}-4\right)-8\left(2x-3\right)=40$$

Use the distributive property to multiply 5 by x^{2}-4.

$$5x^{2}-20-8\left(2x-3\right)=40$$

Use the distributive property to multiply -8 by 2x-3.

$$5x^{2}-20-16x+24=40$$

Add -20 and 24 to get 4.

$$5x^{2}+4-16x=40$$

Subtract 40 from both sides.

$$5x^{2}+4-16x-40=0$$

Subtract 40 from 4 to get -36.

$$5x^{2}-36-16x=0$$

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.

$$5x^{2}-16x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -16 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 5\left(-36\right)}}{2\times 5}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 5\left(-36\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-16\right)±\sqrt{256-20\left(-36\right)}}{2\times 5}$$

Multiply -20 times -36.

$$x=\frac{-\left(-16\right)±\sqrt{256+720}}{2\times 5}$$

Add 256 to 720.

$$x=\frac{-\left(-16\right)±\sqrt{976}}{2\times 5}$$

Take the square root of 976.

$$x=\frac{-\left(-16\right)±4\sqrt{61}}{2\times 5}$$

The opposite of -16 is 16.

$$x=\frac{16±4\sqrt{61}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{16±4\sqrt{61}}{10}$$

Now solve the equation x=\frac{16±4\sqrt{61}}{10} when ± is plus. Add 16 to 4\sqrt{61}.

$$x=\frac{4\sqrt{61}+16}{10}$$

Divide 16+4\sqrt{61} by 10.

$$x=\frac{2\sqrt{61}+8}{5}$$

Now solve the equation x=\frac{16±4\sqrt{61}}{10} when ± is minus. Subtract 4\sqrt{61} from 16.

$$x=\frac{16-4\sqrt{61}}{10}$$

Divide 16-4\sqrt{61} by 10.

$$x=\frac{8-2\sqrt{61}}{5}$$

The equation is now solved.

$$x=\frac{2\sqrt{61}+8}{5} x=\frac{8-2\sqrt{61}}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 40, the least common multiple of 8,5.

$$5\left(x^{2}-4\right)-8\left(2x-3\right)=40$$

Use the distributive property to multiply 5 by x^{2}-4.

$$5x^{2}-20-8\left(2x-3\right)=40$$

Use the distributive property to multiply -8 by 2x-3.

$$5x^{2}-20-16x+24=40$$

Add -20 and 24 to get 4.

$$5x^{2}+4-16x=40$$

Subtract 4 from both sides.

$$5x^{2}-16x=40-4$$

Subtract 4 from 40 to get 36.

$$5x^{2}-16x=36$$

Divide both sides by 5.

$$\frac{5x^{2}-16x}{5}=\frac{36}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{16}{5}x=\frac{36}{5}$$

Divide -\frac{16}{5}, the coefficient of the x term, by 2 to get -\frac{8}{5}. Then add the square of -\frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{5}x+\left(-\frac{8}{5}\right)^{2}=\frac{36}{5}+\left(-\frac{8}{5}\right)^{2}$$

Square -\frac{8}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=\frac{36}{5}+\frac{64}{25}$$

Add \frac{36}{5} to \frac{64}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=\frac{244}{25}$$

Factor x^{2}-\frac{16}{5}x+\frac{64}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{5}\right)^{2}=\frac{244}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{5}\right)^{2}}=\sqrt{\frac{244}{25}}$$

Simplify.

$$x-\frac{8}{5}=\frac{2\sqrt{61}}{5} x-\frac{8}{5}=-\frac{2\sqrt{61}}{5}$$

Add \frac{8}{5} to both sides of the equation.

$$x=\frac{2\sqrt{61}+8}{5} x=\frac{8-2\sqrt{61}}{5}$$