Solve 77x^2+54x-3=0

$$77 { x }^{ 2 } +54x-3=0$$

Solve for x

$$x=\frac{8\sqrt{15}-27}{77}\approx 0.051738529$$

$$x=\frac{-8\sqrt{15}-27}{77}\approx -0.753037231$$

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.

$$77x^{2}+54x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 77 for a, 54 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\times 77\left(-3\right)}}{2\times 77}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\times 77\left(-3\right)}}{2\times 77}$$

Multiply -4 times 77.

$$x=\frac{-54±\sqrt{2916-308\left(-3\right)}}{2\times 77}$$

Multiply -308 times -3.

$$x=\frac{-54±\sqrt{2916+924}}{2\times 77}$$

Add 2916 to 924.

$$x=\frac{-54±\sqrt{3840}}{2\times 77}$$

Take the square root of 3840.

$$x=\frac{-54±16\sqrt{15}}{2\times 77}$$

Multiply 2 times 77.

$$x=\frac{-54±16\sqrt{15}}{154}$$

Now solve the equation x=\frac{-54±16\sqrt{15}}{154} when ± is plus. Add -54 to 16\sqrt{15}.

$$x=\frac{16\sqrt{15}-54}{154}$$

Divide -54+16\sqrt{15} by 154.

$$x=\frac{8\sqrt{15}-27}{77}$$

Now solve the equation x=\frac{-54±16\sqrt{15}}{154} when ± is minus. Subtract 16\sqrt{15} from -54.

$$x=\frac{-16\sqrt{15}-54}{154}$$

Divide -54-16\sqrt{15} by 154.

$$x=\frac{-8\sqrt{15}-27}{77}$$

The equation is now solved.

$$x=\frac{8\sqrt{15}-27}{77} x=\frac{-8\sqrt{15}-27}{77}$$

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.

$$77x^{2}+54x-3=0$$

Add 3 to both sides of the equation.

$$77x^{2}+54x-3-\left(-3\right)=-\left(-3\right)$$

Subtracting -3 from itself leaves 0.

$$77x^{2}+54x=-\left(-3\right)$$

Subtract -3 from 0.

$$77x^{2}+54x=3$$

Divide both sides by 77.

$$\frac{77x^{2}+54x}{77}=\frac{3}{77}$$

Dividing by 77 undoes the multiplication by 77.

$$x^{2}+\frac{54}{77}x=\frac{3}{77}$$

Divide \frac{54}{77}, the coefficient of the x term, by 2 to get \frac{27}{77}. Then add the square of \frac{27}{77} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{54}{77}x+\left(\frac{27}{77}\right)^{2}=\frac{3}{77}+\left(\frac{27}{77}\right)^{2}$$

Square \frac{27}{77} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{54}{77}x+\frac{729}{5929}=\frac{3}{77}+\frac{729}{5929}$$

Add \frac{3}{77} to \frac{729}{5929} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{54}{77}x+\frac{729}{5929}=\frac{960}{5929}$$

Factor x^{2}+\frac{54}{77}x+\frac{729}{5929}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{27}{77}\right)^{2}=\frac{960}{5929}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{77}\right)^{2}}=\sqrt{\frac{960}{5929}}$$

Simplify.

$$x+\frac{27}{77}=\frac{8\sqrt{15}}{77} x+\frac{27}{77}=-\frac{8\sqrt{15}}{77}$$

Subtract \frac{27}{77} from both sides of the equation.

$$x=\frac{8\sqrt{15}-27}{77} x=\frac{-8\sqrt{15}-27}{77}$$