Solve 14c^2+56c=210

$$14 c ^ { 2 } + 56 c = 210$$

Solve for c

$$c=\sqrt{19}-2\approx 2.358898944$$

$$c=-\sqrt{19}-2\approx -6.358898944$$

Steps Using the Quadratic Formula

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.

$$14c^{2}+56c=210$$

Subtract 210 from both sides of the equation.

$$14c^{2}+56c-210=210-210$$

Subtracting 210 from itself leaves 0.

$$14c^{2}+56c-210=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, 56 for b, and -210 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-56±\sqrt{56^{2}-4\times 14\left(-210\right)}}{2\times 14}$$

Square 56.

$$c=\frac{-56±\sqrt{3136-4\times 14\left(-210\right)}}{2\times 14}$$

Multiply -4 times 14.

$$c=\frac{-56±\sqrt{3136-56\left(-210\right)}}{2\times 14}$$

Multiply -56 times -210.

$$c=\frac{-56±\sqrt{3136+11760}}{2\times 14}$$

Add 3136 to 11760.

$$c=\frac{-56±\sqrt{14896}}{2\times 14}$$

Take the square root of 14896.

$$c=\frac{-56±28\sqrt{19}}{2\times 14}$$

Multiply 2 times 14.

$$c=\frac{-56±28\sqrt{19}}{28}$$

Now solve the equation c=\frac{-56±28\sqrt{19}}{28} when ± is plus. Add -56 to 28\sqrt{19}.

$$c=\frac{28\sqrt{19}-56}{28}$$

Divide -56+28\sqrt{19} by 28.

$$c=\sqrt{19}-2$$

Now solve the equation c=\frac{-56±28\sqrt{19}}{28} when ± is minus. Subtract 28\sqrt{19} from -56.

$$c=\frac{-28\sqrt{19}-56}{28}$$

Divide -56-28\sqrt{19} by 28.

$$c=-\sqrt{19}-2$$

The equation is now solved.

$$c=\sqrt{19}-2 c=-\sqrt{19}-2$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$14c^{2}+56c=210$$

Divide both sides by 14.

$$\frac{14c^{2}+56c}{14}=\frac{210}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$c^{2}+\frac{56}{14}c=\frac{210}{14}$$

Divide 56 by 14.

$$c^{2}+4c=\frac{210}{14}$$

Divide 210 by 14.

$$c^{2}+4c=15$$

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.

$$c^{2}+4c+2^{2}=15+2^{2}$$

Square 2.

$$c^{2}+4c+4=15+4$$

Add 15 to 4.

$$c^{2}+4c+4=19$$

Factor c^{2}+4c+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}.

$$\left(c+2\right)^{2}=19$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+2\right)^{2}}=\sqrt{19}$$

Simplify.

$$c+2=\sqrt{19} c+2=-\sqrt{19}$$

Subtract 2 from both sides of the equation.

$$c=\sqrt{19}-2 c=-\sqrt{19}-2$$