Solve 36-9x+-3x^2=0

$$36-9x+-3 { x }^{ 2 } = 0$$

Solve for x

$$x = \frac{\sqrt{57} - 3}{2} \approx 2.274917218$$

$$x=\frac{-\sqrt{57}-3}{2}\approx -5.274917218$$

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.

$$-3x^{2}-9x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -9 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-3\right)\times 36}}{2\left(-3\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-3\right)\times 36}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-9\right)±\sqrt{81+12\times 36}}{2\left(-3\right)}$$

Multiply 12 times 36.

$$x=\frac{-\left(-9\right)±\sqrt{81+432}}{2\left(-3\right)}$$

Add 81 to 432.

$$x=\frac{-\left(-9\right)±\sqrt{513}}{2\left(-3\right)}$$

Take the square root of 513.

$$x=\frac{-\left(-9\right)±3\sqrt{57}}{2\left(-3\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±3\sqrt{57}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{9±3\sqrt{57}}{-6}$$

Now solve the equation x=\frac{9±3\sqrt{57}}{-6} when ± is plus. Add 9 to 3\sqrt{57}.

$$x=\frac{3\sqrt{57}+9}{-6}$$

Divide 9+3\sqrt{57} by -6.

$$x=\frac{-\sqrt{57}-3}{2}$$

Now solve the equation x=\frac{9±3\sqrt{57}}{-6} when ± is minus. Subtract 3\sqrt{57} from 9.

$$x=\frac{9-3\sqrt{57}}{-6}$$

Divide 9-3\sqrt{57} by -6.

$$x=\frac{\sqrt{57}-3}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{57}-3}{2} x=\frac{\sqrt{57}-3}{2}$$

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.

$$-3x^{2}-9x+36=0$$

Subtract 36 from both sides of the equation.

$$-3x^{2}-9x+36-36=-36$$

Subtracting 36 from itself leaves 0.

$$-3x^{2}-9x=-36$$

Divide both sides by -3.

$$\frac{-3x^{2}-9x}{-3}=-\frac{36}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{9}{-3}\right)x=-\frac{36}{-3}$$

Divide -9 by -3.

$$x^{2}+3x=-\frac{36}{-3}$$

Divide -36 by -3.

$$x^{2}+3x=12$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=12+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=12+\frac{9}{4}$$

Add 12 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{57}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{57}}{2} x+\frac{3}{2}=-\frac{\sqrt{57}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{57}-3}{2} x=\frac{-\sqrt{57}-3}{2}$$