Solve u^2+10u-14u-140=-104

$$u ^ { 2 } + 10 u - 14 u - 140 = - 104$$

Solve for u

$$u=2\sqrt{10}+2\approx 8.32455532$$

$$u=2-2\sqrt{10}\approx -4.32455532$$

Steps Using the Quadratic Formula

Combine 10u and -14u to get -4u.

$$u^{2}-4u-140=-104$$

Add 104 to both sides.

$$u^{2}-4u-140+104=0$$

Add -140 and 104 to get -36.

$$u^{2}-4u-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-36\right)}}{2}$$

Square -4.

$$u=\frac{-\left(-4\right)±\sqrt{16-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$u=\frac{-\left(-4\right)±\sqrt{16+144}}{2}$$

Add 16 to 144.

$$u=\frac{-\left(-4\right)±\sqrt{160}}{2}$$

Take the square root of 160.

$$u=\frac{-\left(-4\right)±4\sqrt{10}}{2}$$

The opposite of -4 is 4.

$$u=\frac{4±4\sqrt{10}}{2}$$

Now solve the equation u=\frac{4±4\sqrt{10}}{2} when ± is plus. Add 4 to 4\sqrt{10}.

$$u=\frac{4\sqrt{10}+4}{2}$$

Divide 4+4\sqrt{10} by 2.

$$u=2\sqrt{10}+2$$

Now solve the equation u=\frac{4±4\sqrt{10}}{2} when ± is minus. Subtract 4\sqrt{10} from 4.

$$u=\frac{4-4\sqrt{10}}{2}$$

Divide 4-4\sqrt{10} by 2.

$$u=2-2\sqrt{10}$$

The equation is now solved.

$$u=2\sqrt{10}+2 u=2-2\sqrt{10}$$

Steps for Completing the Square

Combine 10u and -14u to get -4u.

$$u^{2}-4u-140=-104$$

Add 140 to both sides.

$$u^{2}-4u=-104+140$$

Add -104 and 140 to get 36.

$$u^{2}-4u=36$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}-4u+\left(-2\right)^{2}=36+\left(-2\right)^{2}$$

Square -2.

$$u^{2}-4u+4=36+4$$

Add 36 to 4.

$$u^{2}-4u+4=40$$

Factor u^{2}-4u+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-2\right)^{2}=40$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-2\right)^{2}}=\sqrt{40}$$

Simplify.

$$u-2=2\sqrt{10} u-2=-2\sqrt{10}$$

Add 2 to both sides of the equation.

$$u=2\sqrt{10}+2 u=2-2\sqrt{10}$$