Solve (c+7)^2-4(1+7)+3=0

$$( c + 7 ) ^ { 2 } - 4 ( 1 + 7 ) + 3 = 0$$

Solve for c

$$c=\sqrt{29}-7\approx -1.614835193$$

$$c=-\left(\sqrt{29}+7\right)\approx -12.385164807$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(c+7\right)^{2}.

$$c^{2}+14c+49-4\left(1+7\right)+3=0$$

Add 1 and 7 to get 8.

$$c^{2}+14c+49-4\times 8+3=0$$

Multiply 4 and 8 to get 32.

$$c^{2}+14c+49-32+3=0$$

Subtract 32 from 49 to get 17.

$$c^{2}+14c+17+3=0$$

Add 17 and 3 to get 20.

$$c^{2}+14c+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-14±\sqrt{14^{2}-4\times 20}}{2}$$

Square 14.

$$c=\frac{-14±\sqrt{196-4\times 20}}{2}$$

Multiply -4 times 20.

$$c=\frac{-14±\sqrt{196-80}}{2}$$

Add 196 to -80.

$$c=\frac{-14±\sqrt{116}}{2}$$

Take the square root of 116.

$$c=\frac{-14±2\sqrt{29}}{2}$$

Now solve the equation c=\frac{-14±2\sqrt{29}}{2} when ± is plus. Add -14 to 2\sqrt{29}.

$$c=\frac{2\sqrt{29}-14}{2}$$

Divide -14+2\sqrt{29} by 2.

$$c=\sqrt{29}-7$$

Now solve the equation c=\frac{-14±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from -14.

$$c=\frac{-2\sqrt{29}-14}{2}$$

Divide -14-2\sqrt{29} by 2.

$$c=-\sqrt{29}-7$$

The equation is now solved.

$$c=\sqrt{29}-7 c=-\sqrt{29}-7$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(c+7\right)^{2}.

$$c^{2}+14c+49-4\left(1+7\right)+3=0$$

Add 1 and 7 to get 8.

$$c^{2}+14c+49-4\times 8+3=0$$

Multiply 4 and 8 to get 32.

$$c^{2}+14c+49-32+3=0$$

Subtract 32 from 49 to get 17.

$$c^{2}+14c+17+3=0$$

Add 17 and 3 to get 20.

$$c^{2}+14c+20=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$c^{2}+14c=-20$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}+14c+7^{2}=-20+7^{2}$$

Square 7.

$$c^{2}+14c+49=-20+49$$

Add -20 to 49.

$$c^{2}+14c+49=29$$

Factor c^{2}+14c+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c+7\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+7\right)^{2}}=\sqrt{29}$$

Simplify.

$$c+7=\sqrt{29} c+7=-\sqrt{29}$$

Subtract 7 from both sides of the equation.

$$c=\sqrt{29}-7 c=-\sqrt{29}-7$$