Solve 4x^2+48x-x^2-76-1=0

$$4 x ^ { 2 } + 48 x - x ^ { 2 } - 76 - 1 = 0$$

Solve for x

$$x=\frac{\sqrt{807}}{3}-8\approx 1.469248474$$

$$x=-\frac{\sqrt{807}}{3}-8\approx -17.469248474$$

Steps Using the Quadratic Formula

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+48x-76-1=0$$

Subtract 1 from -76 to get -77.

$$3x^{2}+48x-77=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 48 for b, and -77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 3\left(-77\right)}}{2\times 3}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\times 3\left(-77\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-48±\sqrt{2304-12\left(-77\right)}}{2\times 3}$$

Multiply -12 times -77.

$$x=\frac{-48±\sqrt{2304+924}}{2\times 3}$$

Add 2304 to 924.

$$x=\frac{-48±\sqrt{3228}}{2\times 3}$$

Take the square root of 3228.

$$x=\frac{-48±2\sqrt{807}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-48±2\sqrt{807}}{6}$$

Now solve the equation x=\frac{-48±2\sqrt{807}}{6} when ± is plus. Add -48 to 2\sqrt{807}.

$$x=\frac{2\sqrt{807}-48}{6}$$

Divide -48+2\sqrt{807} by 6.

$$x=\frac{\sqrt{807}}{3}-8$$

Now solve the equation x=\frac{-48±2\sqrt{807}}{6} when ± is minus. Subtract 2\sqrt{807} from -48.

$$x=\frac{-2\sqrt{807}-48}{6}$$

Divide -48-2\sqrt{807} by 6.

$$x=-\frac{\sqrt{807}}{3}-8$$

The equation is now solved.

$$x=\frac{\sqrt{807}}{3}-8 x=-\frac{\sqrt{807}}{3}-8$$

Steps for Completing the Square

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+48x-76-1=0$$

Subtract 1 from -76 to get -77.

$$3x^{2}+48x-77=0$$

Add 77 to both sides. Anything plus zero gives itself.

$$3x^{2}+48x=77$$

Divide both sides by 3.

$$\frac{3x^{2}+48x}{3}=\frac{77}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{48}{3}x=\frac{77}{3}$$

Divide 48 by 3.

$$x^{2}+16x=\frac{77}{3}$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=\frac{77}{3}+8^{2}$$

Square 8.

$$x^{2}+16x+64=\frac{77}{3}+64$$

Add \frac{77}{3} to 64.

$$x^{2}+16x+64=\frac{269}{3}$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=\frac{269}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{\frac{269}{3}}$$

Simplify.

$$x+8=\frac{\sqrt{807}}{3} x+8=-\frac{\sqrt{807}}{3}$$

Subtract 8 from both sides of the equation.

$$x=\frac{\sqrt{807}}{3}-8 x=-\frac{\sqrt{807}}{3}-8$$