Solve 48x+54x^2-772=0

$$48x+54 { x }^{ 2 } -772 = 0$$

Solve for x

$$x = \frac{\sqrt{1174} - 4}{9} \approx 3.362631489$$

$$x=\frac{-\sqrt{1174}-4}{9}\approx -4.251520378$$

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.

$$54x^{2}+48x-772=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 54 for a, 48 for b, and -772 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 54\left(-772\right)}}{2\times 54}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 54\left(-772\right)}}{2\times 54}$$

Multiply -4 times 54.

$$x=\frac{-48±\sqrt{2304-216\left(-772\right)}}{2\times 54}$$

Multiply -216 times -772.

$$x=\frac{-48±\sqrt{2304+166752}}{2\times 54}$$

Add 2304 to 166752.

$$x=\frac{-48±\sqrt{169056}}{2\times 54}$$

Take the square root of 169056.

$$x=\frac{-48±12\sqrt{1174}}{2\times 54}$$

Multiply 2 times 54.

$$x=\frac{-48±12\sqrt{1174}}{108}$$

Now solve the equation x=\frac{-48±12\sqrt{1174}}{108} when ± is plus. Add -48 to 12\sqrt{1174}.

$$x=\frac{12\sqrt{1174}-48}{108}$$

Divide -48+12\sqrt{1174} by 108.

$$x=\frac{\sqrt{1174}-4}{9}$$

Now solve the equation x=\frac{-48±12\sqrt{1174}}{108} when ± is minus. Subtract 12\sqrt{1174} from -48.

$$x=\frac{-12\sqrt{1174}-48}{108}$$

Divide -48-12\sqrt{1174} by 108.

$$x=\frac{-\sqrt{1174}-4}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{1174}-4}{9} x=\frac{-\sqrt{1174}-4}{9}$$

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.

$$54x^{2}+48x-772=0$$

Add 772 to both sides of the equation.

$$54x^{2}+48x-772-\left(-772\right)=-\left(-772\right)$$

Subtracting -772 from itself leaves 0.

$$54x^{2}+48x=-\left(-772\right)$$

Subtract -772 from 0.

$$54x^{2}+48x=772$$

Divide both sides by 54.

$$\frac{54x^{2}+48x}{54}=\frac{772}{54}$$

Dividing by 54 undoes the multiplication by 54.

$$x^{2}+\frac{48}{54}x=\frac{772}{54}$$

Reduce the fraction \frac{48}{54} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{8}{9}x=\frac{772}{54}$$

Reduce the fraction \frac{772}{54} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{8}{9}x=\frac{386}{27}$$

Divide \frac{8}{9}, the coefficient of the x term, by 2 to get \frac{4}{9}. Then add the square of \frac{4}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{9}x+\left(\frac{4}{9}\right)^{2}=\frac{386}{27}+\left(\frac{4}{9}\right)^{2}$$

Square \frac{4}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=\frac{386}{27}+\frac{16}{81}$$

Add \frac{386}{27} to \frac{16}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=\frac{1174}{81}$$

Factor x^{2}+\frac{8}{9}x+\frac{16}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4}{9}\right)^{2}=\frac{1174}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{9}\right)^{2}}=\sqrt{\frac{1174}{81}}$$

Simplify.

$$x+\frac{4}{9}=\frac{\sqrt{1174}}{9} x+\frac{4}{9}=-\frac{\sqrt{1174}}{9}$$

Subtract \frac{4}{9} from both sides of the equation.

$$x=\frac{\sqrt{1174}-4}{9} x=\frac{-\sqrt{1174}-4}{9}$$