Solve -0.6x^2+21.6x-25=169.4

$$-0.6 { x }^{ 2 } +21.6x-25=169.4$$

Solve for x

$$x=18$$

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.

$$-0.6x^{2}+21.6x-25=169.4$$

Subtract 169.4 from both sides of the equation.

$$-0.6x^{2}+21.6x-25-169.4=169.4-169.4$$

Subtracting 169.4 from itself leaves 0.

$$-0.6x^{2}+21.6x-25-169.4=0$$

Subtract 169.4 from -25.

$$-0.6x^{2}+21.6x-194.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.6 for a, 21.6 for b, and -194.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21.6±\sqrt{21.6^{2}-4\left(-0.6\right)\left(-194.4\right)}}{2\left(-0.6\right)}$$

Square 21.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-21.6±\sqrt{466.56-4\left(-0.6\right)\left(-194.4\right)}}{2\left(-0.6\right)}$$

Multiply -4 times -0.6.

$$x=\frac{-21.6±\sqrt{466.56+2.4\left(-194.4\right)}}{2\left(-0.6\right)}$$

Multiply 2.4 times -194.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-21.6±\sqrt{\frac{11664-11664}{25}}}{2\left(-0.6\right)}$$

Add 466.56 to -466.56 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-21.6±\sqrt{0}}{2\left(-0.6\right)}$$

Take the square root of 0.

$$x=-\frac{21.6}{2\left(-0.6\right)}$$

Multiply 2 times -0.6.

$$x=-\frac{21.6}{-1.2}$$

Divide -21.6 by -1.2 by multiplying -21.6 by the reciprocal of -1.2.

$$x=18$$

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.

$$-0.6x^{2}+21.6x-25=169.4$$

Add 25 to both sides of the equation.

$$-0.6x^{2}+21.6x-25-\left(-25\right)=169.4-\left(-25\right)$$

Subtracting -25 from itself leaves 0.

$$-0.6x^{2}+21.6x=169.4-\left(-25\right)$$

Subtract -25 from 169.4.

$$-0.6x^{2}+21.6x=194.4$$

Divide both sides of the equation by -0.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.6x^{2}+21.6x}{-0.6}=\frac{194.4}{-0.6}$$

Dividing by -0.6 undoes the multiplication by -0.6.

$$x^{2}+\frac{21.6}{-0.6}x=\frac{194.4}{-0.6}$$

Divide 21.6 by -0.6 by multiplying 21.6 by the reciprocal of -0.6.

$$x^{2}-36x=\frac{194.4}{-0.6}$$

Divide 194.4 by -0.6 by multiplying 194.4 by the reciprocal of -0.6.

$$x^{2}-36x=-324$$

Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-36x+\left(-18\right)^{2}=-324+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-324+324$$

Add -324 to 324.

$$x^{2}-36x+324=0$$

Factor x^{2}-36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-18\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-18=0 x-18=0$$

Add 18 to both sides of the equation.

$$x=18 x=18$$

The equation is now solved. Solutions are the same.

$$x=18$$