Solve x^2+19x^2+56x-196=0

$$x ^ { 2 } + 19 x ^ { 2 } + 56 x - 196 = 0$$

Solve for x

$$x = \frac{7 \sqrt{6} - 7}{5} \approx 2.02928564$$

$$x=\frac{-7\sqrt{6}-7}{5}\approx -4.82928564$$

Steps Using the Quadratic Formula

Combine x^{2} and 19x^{2} to get 20x^{2}.

$$20x^{2}+56x-196=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, 56 for b, and -196 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\times 20\left(-196\right)}}{2\times 20}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\times 20\left(-196\right)}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-56±\sqrt{3136-80\left(-196\right)}}{2\times 20}$$

Multiply -80 times -196.

$$x=\frac{-56±\sqrt{3136+15680}}{2\times 20}$$

Add 3136 to 15680.

$$x=\frac{-56±\sqrt{18816}}{2\times 20}$$

Take the square root of 18816.

$$x=\frac{-56±56\sqrt{6}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{-56±56\sqrt{6}}{40}$$

Now solve the equation x=\frac{-56±56\sqrt{6}}{40} when ± is plus. Add -56 to 56\sqrt{6}.

$$x=\frac{56\sqrt{6}-56}{40}$$

Divide -56+56\sqrt{6} by 40.

$$x=\frac{7\sqrt{6}-7}{5}$$

Now solve the equation x=\frac{-56±56\sqrt{6}}{40} when ± is minus. Subtract 56\sqrt{6} from -56.

$$x=\frac{-56\sqrt{6}-56}{40}$$

Divide -56-56\sqrt{6} by 40.

$$x=\frac{-7\sqrt{6}-7}{5}$$

The equation is now solved.

$$x=\frac{7\sqrt{6}-7}{5} x=\frac{-7\sqrt{6}-7}{5}$$

Steps for Completing the Square

Combine x^{2} and 19x^{2} to get 20x^{2}.

$$20x^{2}+56x-196=0$$

Add 196 to both sides. Anything plus zero gives itself.

$$20x^{2}+56x=196$$

Divide both sides by 20.

$$\frac{20x^{2}+56x}{20}=\frac{196}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}+\frac{56}{20}x=\frac{196}{20}$$

Reduce the fraction \frac{56}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{14}{5}x=\frac{196}{20}$$

Reduce the fraction \frac{196}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{14}{5}x=\frac{49}{5}$$

Divide \frac{14}{5}, the coefficient of the x term, by 2 to get \frac{7}{5}. Then add the square of \frac{7}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{14}{5}x+\left(\frac{7}{5}\right)^{2}=\frac{49}{5}+\left(\frac{7}{5}\right)^{2}$$

Square \frac{7}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{49}{5}+\frac{49}{25}$$

Add \frac{49}{5} to \frac{49}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{294}{25}$$

Factor x^{2}+\frac{14}{5}x+\frac{49}{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}.

$$\left(x+\frac{7}{5}\right)^{2}=\frac{294}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{5}\right)^{2}}=\sqrt{\frac{294}{25}}$$

Simplify.

$$x+\frac{7}{5}=\frac{7\sqrt{6}}{5} x+\frac{7}{5}=-\frac{7\sqrt{6}}{5}$$

Subtract \frac{7}{5} from both sides of the equation.

$$x=\frac{7\sqrt{6}-7}{5} x=\frac{-7\sqrt{6}-7}{5}$$