Solve 12x^2-552x+4320=0

$$12 { x }^{ 2 } -552x+4320=0$$

Solve for x

$$x=10$$

$$x=36$$

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.

$$12x^{2}-552x+4320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -552 for b, and 4320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-552\right)±\sqrt{\left(-552\right)^{2}-4\times 12\times 4320}}{2\times 12}$$

Square -552.

$$x=\frac{-\left(-552\right)±\sqrt{304704-4\times 12\times 4320}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-552\right)±\sqrt{304704-48\times 4320}}{2\times 12}$$

Multiply -48 times 4320.

$$x=\frac{-\left(-552\right)±\sqrt{304704-207360}}{2\times 12}$$

Add 304704 to -207360.

$$x=\frac{-\left(-552\right)±\sqrt{97344}}{2\times 12}$$

Take the square root of 97344.

$$x=\frac{-\left(-552\right)±312}{2\times 12}$$

The opposite of -552 is 552.

$$x=\frac{552±312}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{552±312}{24}$$

Now solve the equation x=\frac{552±312}{24} when ± is plus. Add 552 to 312.

$$x=\frac{864}{24}$$

Divide 864 by 24.

$$x=36$$

Now solve the equation x=\frac{552±312}{24} when ± is minus. Subtract 312 from 552.

$$x=\frac{240}{24}$$

Divide 240 by 24.

$$x=10$$

The equation is now solved.

$$x=36 x=10$$

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.

$$12x^{2}-552x+4320=0$$

Subtract 4320 from both sides of the equation.

$$12x^{2}-552x+4320-4320=-4320$$

Subtracting 4320 from itself leaves 0.

$$12x^{2}-552x=-4320$$

Divide both sides by 12.

$$\frac{12x^{2}-552x}{12}=-\frac{4320}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{552}{12}\right)x=-\frac{4320}{12}$$

Divide -552 by 12.

$$x^{2}-46x=-\frac{4320}{12}$$

Divide -4320 by 12.

$$x^{2}-46x=-360$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-360+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-360+529$$

Add -360 to 529.

$$x^{2}-46x+529=169$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{169}$$

Simplify.

$$x-23=13 x-23=-13$$

Add 23 to both sides of the equation.

$$x=36 x=10$$