Solve 7.2x^2-72x+225=0

$$7.2 x ^ { 2 } - 72 x + 225 = 0$$

Solve for x (complex solution)

$$x=5+2.5i$$

$$x=5-2.5i$$

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.

$$7.2x^{2}-72x+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7.2 for a, -72 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 7.2\times 225}}{2\times 7.2}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 7.2\times 225}}{2\times 7.2}$$

Multiply -4 times 7.2.

$$x=\frac{-\left(-72\right)±\sqrt{5184-28.8\times 225}}{2\times 7.2}$$

Multiply -28.8 times 225.

$$x=\frac{-\left(-72\right)±\sqrt{5184-6480}}{2\times 7.2}$$

Add 5184 to -6480.

$$x=\frac{-\left(-72\right)±\sqrt{-1296}}{2\times 7.2}$$

Take the square root of -1296.

$$x=\frac{-\left(-72\right)±36i}{2\times 7.2}$$

The opposite of -72 is 72.

$$x=\frac{72±36i}{2\times 7.2}$$

Multiply 2 times 7.2.

$$x=\frac{72±36i}{14.4}$$

Now solve the equation x=\frac{72±36i}{14.4} when ± is plus. Add 72 to 36i.

$$x=\frac{72+36i}{14.4}$$

Divide 72+36i by 14.4 by multiplying 72+36i by the reciprocal of 14.4.

$$x=5+2.5i$$

Now solve the equation x=\frac{72±36i}{14.4} when ± is minus. Subtract 36i from 72.

$$x=\frac{72-36i}{14.4}$$

Divide 72-36i by 14.4 by multiplying 72-36i by the reciprocal of 14.4.

$$x=5-2.5i$$

The equation is now solved.

$$x=5+2.5i x=5-2.5i$$

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.

$$7.2x^{2}-72x+225=0$$

Subtract 225 from both sides of the equation.

$$7.2x^{2}-72x+225-225=-225$$

Subtracting 225 from itself leaves 0.

$$7.2x^{2}-72x=-225$$

Divide both sides of the equation by 7.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{7.2x^{2}-72x}{7.2}=-\frac{225}{7.2}$$

Dividing by 7.2 undoes the multiplication by 7.2.

$$x^{2}+\left(-\frac{72}{7.2}\right)x=-\frac{225}{7.2}$$

Divide -72 by 7.2 by multiplying -72 by the reciprocal of 7.2.

$$x^{2}-10x=-\frac{225}{7.2}$$

Divide -225 by 7.2 by multiplying -225 by the reciprocal of 7.2.

$$x^{2}-10x=-31.25$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-31.25+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-31.25+25$$

Add -31.25 to 25.

$$x^{2}-10x+25=-6.25$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=-6.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-6.25}$$

Simplify.

$$x-5=\frac{5}{2}i x-5=-\frac{5}{2}i$$

Add 5 to both sides of the equation.

$$x=5+\frac{5}{2}i x=5-\frac{5}{2}i$$