Solve x^2-21x+40.5=0

$${ x }^{ 2 } -21x+40.5=0$$

Solve for x

$$x = \frac{3 \sqrt{31} + 21}{2} \approx 18.851646544$$

$$x = \frac{21 - 3 \sqrt{31}}{2} \approx 2.148353456$$

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.

$$x^{2}-21x+40.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -21 for b, and 40.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 40.5}}{2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 40.5}}{2}$$

Multiply -4 times 40.5.

$$x=\frac{-\left(-21\right)±\sqrt{441-162}}{2}$$

Add 441 to -162.

$$x=\frac{-\left(-21\right)±\sqrt{279}}{2}$$

Take the square root of 279.

$$x=\frac{-\left(-21\right)±3\sqrt{31}}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±3\sqrt{31}}{2}$$

Now solve the equation x=\frac{21±3\sqrt{31}}{2} when ± is plus. Add 21 to 3\sqrt{31}.

$$x=\frac{3\sqrt{31}+21}{2}$$

Now solve the equation x=\frac{21±3\sqrt{31}}{2} when ± is minus. Subtract 3\sqrt{31} from 21.

$$x=\frac{21-3\sqrt{31}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{31}+21}{2} x=\frac{21-3\sqrt{31}}{2}$$

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.

$$x^{2}-21x+40.5=0$$

Subtract 40.5 from both sides of the equation.

$$x^{2}-21x+40.5-40.5=-40.5$$

Subtracting 40.5 from itself leaves 0.

$$x^{2}-21x=-40.5$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-40.5+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-40.5+\frac{441}{4}$$

Add -40.5 to \frac{441}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-21x+\frac{441}{4}=\frac{279}{4}$$

Factor x^{2}-21x+\frac{441}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{279}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{279}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{3\sqrt{31}}{2} x-\frac{21}{2}=-\frac{3\sqrt{31}}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=\frac{3\sqrt{31}+21}{2} x=\frac{21-3\sqrt{31}}{2}$$