Solve F(x)=x^2-3x^2-45x+50

$$F ( x ) = x ^ { 2 } - 3 x ^ { 2 } - 45 x + 50$$

Factor

$$-2\left(x-\frac{-5\sqrt{97}-45}{4}\right)\left(x-\frac{5\sqrt{97}-45}{4}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-2x^{2}-45x+50)$$

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.

$$-2x^{2}-45x+50=0$$

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.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-45\right)±\sqrt{2025+8\times 50}}{2\left(-2\right)}$$

Multiply 8 times 50.

$$x=\frac{-\left(-45\right)±\sqrt{2025+400}}{2\left(-2\right)}$$

Add 2025 to 400.

$$x=\frac{-\left(-45\right)±\sqrt{2425}}{2\left(-2\right)}$$

Take the square root of 2425.

$$x=\frac{-\left(-45\right)±5\sqrt{97}}{2\left(-2\right)}$$

The opposite of -45 is 45.

$$x=\frac{45±5\sqrt{97}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{45±5\sqrt{97}}{-4}$$

Now solve the equation x=\frac{45±5\sqrt{97}}{-4} when ± is plus. Add 45 to 5\sqrt{97}.

$$x=\frac{5\sqrt{97}+45}{-4}$$

Divide 45+5\sqrt{97} by -4.

$$x=\frac{-5\sqrt{97}-45}{4}$$

Now solve the equation x=\frac{45±5\sqrt{97}}{-4} when ± is minus. Subtract 5\sqrt{97} from 45.

$$x=\frac{45-5\sqrt{97}}{-4}$$

Divide 45-5\sqrt{97} by -4.

$$x=\frac{5\sqrt{97}-45}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-45-5\sqrt{97}}{4} for x_{1} and \frac{-45+5\sqrt{97}}{4} for x_{2}.

$$-2x^{2}-45x+50=-2\left(x-\frac{-5\sqrt{97}-45}{4}\right)\left(x-\frac{5\sqrt{97}-45}{4}\right)$$

Evaluate

$$50-45x-2x^{2}$$

Solution Steps

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

$$-2x^{2}-45x+50$$