Solve x^2+5x-14quadtext{2}quad3x^2+20x+25

$$x ^ { 2 } + 5 x - 14 \quad \text { 2 } \quad 3 x ^ { 2 } + 20 x + 25$$

Evaluate

$$25+25x-83x^{2}$$

Solution Steps

Multiply 14 and 2 to get 28.

$$x^{2}+5x-28\times 3x^{2}+20x+25$$

Multiply 28 and 3 to get 84.

$$x^{2}+5x-84x^{2}+20x+25$$

Combine x^{2} and -84x^{2} to get -83x^{2}.

$$-83x^{2}+5x+20x+25$$

Combine 5x and 20x to get 25x.

$$-83x^{2}+25x+25$$

Factor

$$-83\left(x-\frac{25-5\sqrt{357}}{166}\right)\left(x-\frac{5\sqrt{357}+25}{166}\right)$$

Steps Using the Quadratic Formula

Multiply 14 and 2 to get 28.

$$factor(x^{2}+5x-28\times 3x^{2}+20x+25)$$

Multiply 28 and 3 to get 84.

$$factor(x^{2}+5x-84x^{2}+20x+25)$$

Combine x^{2} and -84x^{2} to get -83x^{2}.

$$factor(-83x^{2}+5x+20x+25)$$

Combine 5x and 20x to get 25x.

$$factor(-83x^{2}+25x+25)$$

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.

$$-83x^{2}+25x+25=0$$

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.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-83\right)\times 25}}{2\left(-83\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-83\right)\times 25}}{2\left(-83\right)}$$

Multiply -4 times -83.

$$x=\frac{-25±\sqrt{625+332\times 25}}{2\left(-83\right)}$$

Multiply 332 times 25.

$$x=\frac{-25±\sqrt{625+8300}}{2\left(-83\right)}$$

Add 625 to 8300.

$$x=\frac{-25±\sqrt{8925}}{2\left(-83\right)}$$

Take the square root of 8925.

$$x=\frac{-25±5\sqrt{357}}{2\left(-83\right)}$$

Multiply 2 times -83.

$$x=\frac{-25±5\sqrt{357}}{-166}$$

Now solve the equation x=\frac{-25±5\sqrt{357}}{-166} when ± is plus. Add -25 to 5\sqrt{357}.

$$x=\frac{5\sqrt{357}-25}{-166}$$

Divide -25+5\sqrt{357} by -166.

$$x=\frac{25-5\sqrt{357}}{166}$$

Now solve the equation x=\frac{-25±5\sqrt{357}}{-166} when ± is minus. Subtract 5\sqrt{357} from -25.

$$x=\frac{-5\sqrt{357}-25}{-166}$$

Divide -25-5\sqrt{357} by -166.

$$x=\frac{5\sqrt{357}+25}{166}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{25-5\sqrt{357}}{166} for x_{1} and \frac{25+5\sqrt{357}}{166} for x_{2}.

$$-83x^{2}+25x+25=-83\left(x-\frac{25-5\sqrt{357}}{166}\right)\left(x-\frac{5\sqrt{357}+25}{166}\right)$$