Solve {l}{-5x^2+7x-14x^2-32x}{-19x^2-25x-1}

$$\left. \begin{array} { l } { - 5 x ^ { 2 } + 7 x - 14 x ^ { 2 } - 32 x } \\ { - 19 x ^ { 2 } - 25 x - 1 } \end{array} \right.$$

Evaluate

$$-38x^{2}-50x-1$$

Solution Steps

Combine -5x^{2} and -14x^{2} to get -19x^{2}.

$$-19x^{2}+7x-32x-19x^{2}-25x-1$$

Combine 7x and -32x to get -25x.

$$-19x^{2}-25x-19x^{2}-25x-1$$

Combine -19x^{2} and -19x^{2} to get -38x^{2}.

$$-38x^{2}-25x-25x-1$$

Combine -25x and -25x to get -50x.

$$-38x^{2}-50x-1$$

Factor

$$-38\left(x-\frac{-\sqrt{587}-25}{38}\right)\left(x-\frac{\sqrt{587}-25}{38}\right)$$

Steps Using the Quadratic Formula

Combine -5x^{2} and -14x^{2} to get -19x^{2}.

$$factor(-19x^{2}+7x-32x-19x^{2}-25x-1)$$

Combine 7x and -32x to get -25x.

$$factor(-19x^{2}-25x-19x^{2}-25x-1)$$

Combine -19x^{2} and -19x^{2} to get -38x^{2}.

$$factor(-38x^{2}-25x-25x-1)$$

Combine -25x and -25x to get -50x.

$$factor(-38x^{2}-50x-1)$$

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.

$$-38x^{2}-50x-1=0$$

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.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-38\right)\left(-1\right)}}{2\left(-38\right)}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-38\right)\left(-1\right)}}{2\left(-38\right)}$$

Multiply -4 times -38.

$$x=\frac{-\left(-50\right)±\sqrt{2500+152\left(-1\right)}}{2\left(-38\right)}$$

Multiply 152 times -1.

$$x=\frac{-\left(-50\right)±\sqrt{2500-152}}{2\left(-38\right)}$$

Add 2500 to -152.

$$x=\frac{-\left(-50\right)±\sqrt{2348}}{2\left(-38\right)}$$

Take the square root of 2348.

$$x=\frac{-\left(-50\right)±2\sqrt{587}}{2\left(-38\right)}$$

The opposite of -50 is 50.

$$x=\frac{50±2\sqrt{587}}{2\left(-38\right)}$$

Multiply 2 times -38.

$$x=\frac{50±2\sqrt{587}}{-76}$$

Now solve the equation x=\frac{50±2\sqrt{587}}{-76} when ± is plus. Add 50 to 2\sqrt{587}.

$$x=\frac{2\sqrt{587}+50}{-76}$$

Divide 50+2\sqrt{587} by -76.

$$x=\frac{-\sqrt{587}-25}{38}$$

Now solve the equation x=\frac{50±2\sqrt{587}}{-76} when ± is minus. Subtract 2\sqrt{587} from 50.

$$x=\frac{50-2\sqrt{587}}{-76}$$

Divide 50-2\sqrt{587} by -76.

$$x=\frac{\sqrt{587}-25}{38}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-25-\sqrt{587}}{38} for x_{1} and \frac{-25+\sqrt{587}}{38} for x_{2}.

$$-38x^{2}-50x-1=-38\left(x-\frac{-\sqrt{587}-25}{38}\right)\left(x-\frac{\sqrt{587}-25}{38}\right)$$