Solve 2a^2+8a+10=a^2

$$2 a ^ { 2 } + 8 a + 10 = a ^ { 2 }$$

Solve for a

$$a=\sqrt{6}-4\approx -1.550510257$$

$$a=-\sqrt{6}-4\approx -6.449489743$$

Steps Using the Quadratic Formula

Subtract a^{2} from both sides.

$$2a^{2}+8a+10-a^{2}=0$$

Combine 2a^{2} and -a^{2} to get a^{2}.

$$a^{2}+8a+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8±\sqrt{8^{2}-4\times 10}}{2}$$

Square 8.

$$a=\frac{-8±\sqrt{64-4\times 10}}{2}$$

Multiply -4 times 10.

$$a=\frac{-8±\sqrt{64-40}}{2}$$

Add 64 to -40.

$$a=\frac{-8±\sqrt{24}}{2}$$

Take the square root of 24.

$$a=\frac{-8±2\sqrt{6}}{2}$$

Now solve the equation a=\frac{-8±2\sqrt{6}}{2} when ± is plus. Add -8 to 2\sqrt{6}.

$$a=\frac{2\sqrt{6}-8}{2}$$

Divide -8+2\sqrt{6} by 2.

$$a=\sqrt{6}-4$$

Now solve the equation a=\frac{-8±2\sqrt{6}}{2} when ± is minus. Subtract 2\sqrt{6} from -8.

$$a=\frac{-2\sqrt{6}-8}{2}$$

Divide -8-2\sqrt{6} by 2.

$$a=-\sqrt{6}-4$$

The equation is now solved.

$$a=\sqrt{6}-4 a=-\sqrt{6}-4$$

Steps for Completing the Square

Subtract a^{2} from both sides.

$$2a^{2}+8a+10-a^{2}=0$$

Combine 2a^{2} and -a^{2} to get a^{2}.

$$a^{2}+8a+10=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$a^{2}+8a=-10$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+8a+4^{2}=-10+4^{2}$$

Square 4.

$$a^{2}+8a+16=-10+16$$

Add -10 to 16.

$$a^{2}+8a+16=6$$

Factor a^{2}+8a+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+4\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+4\right)^{2}}=\sqrt{6}$$

Simplify.

$$a+4=\sqrt{6} a+4=-\sqrt{6}$$

Subtract 4 from both sides of the equation.

$$a=\sqrt{6}-4 a=-\sqrt{6}-4$$