Solve 2x^2-3x^2-50x+75

$$2 { x }^{ 2 } -3 { x }^{ 2 } -50x+75$$

Factor

$$-\left(x-\left(-10\sqrt{7}-25\right)\right)\left(x-\left(10\sqrt{7}-25\right)\right)$$

Steps Using the Quadratic Formula

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

$$factor(-x^{2}-50x+75)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-x^{2}-50x+75=0$$

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.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-1\right)\times 75}}{2\left(-1\right)}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-1\right)\times 75}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-50\right)±\sqrt{2500+4\times 75}}{2\left(-1\right)}$$

Multiply 4 times 75.

$$x=\frac{-\left(-50\right)±\sqrt{2500+300}}{2\left(-1\right)}$$

Add 2500 to 300.

$$x=\frac{-\left(-50\right)±\sqrt{2800}}{2\left(-1\right)}$$

Take the square root of 2800.

$$x=\frac{-\left(-50\right)±20\sqrt{7}}{2\left(-1\right)}$$

The opposite of -50 is 50.

$$x=\frac{50±20\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{50±20\sqrt{7}}{-2}$$

Now solve the equation x=\frac{50±20\sqrt{7}}{-2} when ± is plus. Add 50 to 20\sqrt{7}.

$$x=\frac{20\sqrt{7}+50}{-2}$$

Divide 50+20\sqrt{7} by -2.

$$x=-10\sqrt{7}-25$$

Now solve the equation x=\frac{50±20\sqrt{7}}{-2} when ± is minus. Subtract 20\sqrt{7} from 50.

$$x=\frac{50-20\sqrt{7}}{-2}$$

Divide 50-20\sqrt{7} by -2.

$$x=10\sqrt{7}-25$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -25-10\sqrt{7} for x_{1} and -25+10\sqrt{7} for x_{2}.

$$-x^{2}-50x+75=-\left(x-\left(-10\sqrt{7}-25\right)\right)\left(x-\left(10\sqrt{7}-25\right)\right)$$

Evaluate

$$75-50x-x^{2}$$

Solution Steps

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

$$-x^{2}-50x+75$$