Solve 3x^2-64x+112=0

$$3 { x }^{ 2 } -64x+112=0$$

Solve for x

$$x = \frac{4 \sqrt{43} + 32}{3} \approx 19.409918032$$

$$x = \frac{32 - 4 \sqrt{43}}{3} \approx 1.923415301$$

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.

$$3x^{2}-64x+112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -64 for b, and 112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 3\times 112}}{2\times 3}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 3\times 112}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-64\right)±\sqrt{4096-12\times 112}}{2\times 3}$$

Multiply -12 times 112.

$$x=\frac{-\left(-64\right)±\sqrt{4096-1344}}{2\times 3}$$

Add 4096 to -1344.

$$x=\frac{-\left(-64\right)±\sqrt{2752}}{2\times 3}$$

Take the square root of 2752.

$$x=\frac{-\left(-64\right)±8\sqrt{43}}{2\times 3}$$

The opposite of -64 is 64.

$$x=\frac{64±8\sqrt{43}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{64±8\sqrt{43}}{6}$$

Now solve the equation x=\frac{64±8\sqrt{43}}{6} when ± is plus. Add 64 to 8\sqrt{43}.

$$x=\frac{8\sqrt{43}+64}{6}$$

Divide 64+8\sqrt{43} by 6.

$$x=\frac{4\sqrt{43}+32}{3}$$

Now solve the equation x=\frac{64±8\sqrt{43}}{6} when ± is minus. Subtract 8\sqrt{43} from 64.

$$x=\frac{64-8\sqrt{43}}{6}$$

Divide 64-8\sqrt{43} by 6.

$$x=\frac{32-4\sqrt{43}}{3}$$

The equation is now solved.

$$x=\frac{4\sqrt{43}+32}{3} x=\frac{32-4\sqrt{43}}{3}$$

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.

$$3x^{2}-64x+112=0$$

Subtract 112 from both sides of the equation.

$$3x^{2}-64x+112-112=-112$$

Subtracting 112 from itself leaves 0.

$$3x^{2}-64x=-112$$

Divide both sides by 3.

$$\frac{3x^{2}-64x}{3}=-\frac{112}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{64}{3}x=-\frac{112}{3}$$

Divide -\frac{64}{3}, the coefficient of the x term, by 2 to get -\frac{32}{3}. Then add the square of -\frac{32}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{64}{3}x+\left(-\frac{32}{3}\right)^{2}=-\frac{112}{3}+\left(-\frac{32}{3}\right)^{2}$$

Square -\frac{32}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{64}{3}x+\frac{1024}{9}=-\frac{112}{3}+\frac{1024}{9}$$

Add -\frac{112}{3} to \frac{1024}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{64}{3}x+\frac{1024}{9}=\frac{688}{9}$$

Factor x^{2}-\frac{64}{3}x+\frac{1024}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32}{3}\right)^{2}=\frac{688}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{32}{3}\right)^{2}}=\sqrt{\frac{688}{9}}$$

Simplify.

$$x-\frac{32}{3}=\frac{4\sqrt{43}}{3} x-\frac{32}{3}=-\frac{4\sqrt{43}}{3}$$

Add \frac{32}{3} to both sides of the equation.

$$x=\frac{4\sqrt{43}+32}{3} x=\frac{32-4\sqrt{43}}{3}$$