Solve -6x^2-30x+450=0

$$-6 { x }^{ 2 } -30x+450 = 0$$

Solve for x

$$x = \frac{5 \sqrt{13} - 5}{2} \approx 6.513878189$$

$$x=\frac{-5\sqrt{13}-5}{2}\approx -11.513878189$$

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.

$$-6x^{2}-30x+450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -30 for b, and 450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-6\right)\times 450}}{2\left(-6\right)}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-6\right)\times 450}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-30\right)±\sqrt{900+24\times 450}}{2\left(-6\right)}$$

Multiply 24 times 450.

$$x=\frac{-\left(-30\right)±\sqrt{900+10800}}{2\left(-6\right)}$$

Add 900 to 10800.

$$x=\frac{-\left(-30\right)±\sqrt{11700}}{2\left(-6\right)}$$

Take the square root of 11700.

$$x=\frac{-\left(-30\right)±30\sqrt{13}}{2\left(-6\right)}$$

The opposite of -30 is 30.

$$x=\frac{30±30\sqrt{13}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{30±30\sqrt{13}}{-12}$$

Now solve the equation x=\frac{30±30\sqrt{13}}{-12} when ± is plus. Add 30 to 30\sqrt{13}.

$$x=\frac{30\sqrt{13}+30}{-12}$$

Divide 30+30\sqrt{13} by -12.

$$x=\frac{-5\sqrt{13}-5}{2}$$

Now solve the equation x=\frac{30±30\sqrt{13}}{-12} when ± is minus. Subtract 30\sqrt{13} from 30.

$$x=\frac{30-30\sqrt{13}}{-12}$$

Divide 30-30\sqrt{13} by -12.

$$x=\frac{5\sqrt{13}-5}{2}$$

The equation is now solved.

$$x=\frac{-5\sqrt{13}-5}{2} x=\frac{5\sqrt{13}-5}{2}$$

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.

$$-6x^{2}-30x+450=0$$

Subtract 450 from both sides of the equation.

$$-6x^{2}-30x+450-450=-450$$

Subtracting 450 from itself leaves 0.

$$-6x^{2}-30x=-450$$

Divide both sides by -6.

$$\frac{-6x^{2}-30x}{-6}=-\frac{450}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\left(-\frac{30}{-6}\right)x=-\frac{450}{-6}$$

Divide -30 by -6.

$$x^{2}+5x=-\frac{450}{-6}$$

Divide -450 by -6.

$$x^{2}+5x=75$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=75+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=75+\frac{25}{4}$$

Add 75 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{325}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{325}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{325}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{5\sqrt{13}}{2} x+\frac{5}{2}=-\frac{5\sqrt{13}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{5\sqrt{13}-5}{2} x=\frac{-5\sqrt{13}-5}{2}$$