Solve 5x^2-1=56x

$$5 x ^ { 2 } - 1 = 56 x$$

Solve for x

$$x = \frac{\sqrt{789} + 28}{5} \approx 11.217828762$$

$$x=\frac{28-\sqrt{789}}{5}\approx -0.017828762$$

Steps Using the Quadratic Formula

Subtract 56x from both sides.

$$5x^{2}-1-56x=0$$

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.

$$5x^{2}-56x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -56 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 5\left(-1\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-56\right)±\sqrt{3136-20\left(-1\right)}}{2\times 5}$$

Multiply -20 times -1.

$$x=\frac{-\left(-56\right)±\sqrt{3136+20}}{2\times 5}$$

Add 3136 to 20.

$$x=\frac{-\left(-56\right)±\sqrt{3156}}{2\times 5}$$

Take the square root of 3156.

$$x=\frac{-\left(-56\right)±2\sqrt{789}}{2\times 5}$$

The opposite of -56 is 56.

$$x=\frac{56±2\sqrt{789}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{56±2\sqrt{789}}{10}$$

Now solve the equation x=\frac{56±2\sqrt{789}}{10} when ± is plus. Add 56 to 2\sqrt{789}.

$$x=\frac{2\sqrt{789}+56}{10}$$

Divide 56+2\sqrt{789} by 10.

$$x=\frac{\sqrt{789}+28}{5}$$

Now solve the equation x=\frac{56±2\sqrt{789}}{10} when ± is minus. Subtract 2\sqrt{789} from 56.

$$x=\frac{56-2\sqrt{789}}{10}$$

Divide 56-2\sqrt{789} by 10.

$$x=\frac{28-\sqrt{789}}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{789}+28}{5} x=\frac{28-\sqrt{789}}{5}$$

Steps for Completing the Square

Subtract 56x from both sides.

$$5x^{2}-1-56x=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$5x^{2}-56x=1$$

Divide both sides by 5.

$$\frac{5x^{2}-56x}{5}=\frac{1}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{56}{5}x=\frac{1}{5}$$

Divide -\frac{56}{5}, the coefficient of the x term, by 2 to get -\frac{28}{5}. Then add the square of -\frac{28}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{56}{5}x+\left(-\frac{28}{5}\right)^{2}=\frac{1}{5}+\left(-\frac{28}{5}\right)^{2}$$

Square -\frac{28}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{56}{5}x+\frac{784}{25}=\frac{1}{5}+\frac{784}{25}$$

Add \frac{1}{5} to \frac{784}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{56}{5}x+\frac{784}{25}=\frac{789}{25}$$

Factor x^{2}-\frac{56}{5}x+\frac{784}{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}.

$$\left(x-\frac{28}{5}\right)^{2}=\frac{789}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{28}{5}\right)^{2}}=\sqrt{\frac{789}{25}}$$

Simplify.

$$x-\frac{28}{5}=\frac{\sqrt{789}}{5} x-\frac{28}{5}=-\frac{\sqrt{789}}{5}$$

Add \frac{28}{5} to both sides of the equation.

$$x=\frac{\sqrt{789}+28}{5} x=\frac{28-\sqrt{789}}{5}$$