Solve 3x^2-6x^2-75x+150=0

$$3 x ^ { 2 } - 6 x ^ { 2 } - 75 x + 150 = 0$$

Solve for x

$$x = \frac{5 \sqrt{33} - 25}{2} \approx 1.861406616$$

$$x=\frac{-5\sqrt{33}-25}{2}\approx -26.861406616$$

Steps Using the Quadratic Formula

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}-75x+150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -75 for b, and 150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-75\right)±\sqrt{\left(-75\right)^{2}-4\left(-3\right)\times 150}}{2\left(-3\right)}$$

Square -75.

$$x=\frac{-\left(-75\right)±\sqrt{5625-4\left(-3\right)\times 150}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-75\right)±\sqrt{5625+12\times 150}}{2\left(-3\right)}$$

Multiply 12 times 150.

$$x=\frac{-\left(-75\right)±\sqrt{5625+1800}}{2\left(-3\right)}$$

Add 5625 to 1800.

$$x=\frac{-\left(-75\right)±\sqrt{7425}}{2\left(-3\right)}$$

Take the square root of 7425.

$$x=\frac{-\left(-75\right)±15\sqrt{33}}{2\left(-3\right)}$$

The opposite of -75 is 75.

$$x=\frac{75±15\sqrt{33}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{75±15\sqrt{33}}{-6}$$

Now solve the equation x=\frac{75±15\sqrt{33}}{-6} when ± is plus. Add 75 to 15\sqrt{33}.

$$x=\frac{15\sqrt{33}+75}{-6}$$

Divide 75+15\sqrt{33} by -6.

$$x=\frac{-5\sqrt{33}-25}{2}$$

Now solve the equation x=\frac{75±15\sqrt{33}}{-6} when ± is minus. Subtract 15\sqrt{33} from 75.

$$x=\frac{75-15\sqrt{33}}{-6}$$

Divide 75-15\sqrt{33} by -6.

$$x=\frac{5\sqrt{33}-25}{2}$$

The equation is now solved.

$$x=\frac{-5\sqrt{33}-25}{2} x=\frac{5\sqrt{33}-25}{2}$$

Steps for Completing the Square

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}-75x+150=0$$

Subtract 150 from both sides. Anything subtracted from zero gives its negation.

$$-3x^{2}-75x=-150$$

Divide both sides by -3.

$$\frac{-3x^{2}-75x}{-3}=-\frac{150}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{75}{-3}\right)x=-\frac{150}{-3}$$

Divide -75 by -3.

$$x^{2}+25x=-\frac{150}{-3}$$

Divide -150 by -3.

$$x^{2}+25x=50$$

Divide 25, the coefficient of the x term, by 2 to get \frac{25}{2}. Then add the square of \frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+25x+\left(\frac{25}{2}\right)^{2}=50+\left(\frac{25}{2}\right)^{2}$$

Square \frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+25x+\frac{625}{4}=50+\frac{625}{4}$$

Add 50 to \frac{625}{4}.

$$x^{2}+25x+\frac{625}{4}=\frac{825}{4}$$

Factor x^{2}+25x+\frac{625}{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}.

$$\left(x+\frac{25}{2}\right)^{2}=\frac{825}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{2}\right)^{2}}=\sqrt{\frac{825}{4}}$$

Simplify.

$$x+\frac{25}{2}=\frac{5\sqrt{33}}{2} x+\frac{25}{2}=-\frac{5\sqrt{33}}{2}$$

Subtract \frac{25}{2} from both sides of the equation.

$$x=\frac{5\sqrt{33}-25}{2} x=\frac{-5\sqrt{33}-25}{2}$$