Solve 5x^2-11x+7-9x-14x^2

$$5 x ^ { 2 } - 11 x + 7 - 9 x - 14 x ^ { 2 }$$

Evaluate

$$7-20x-9x^{2}$$

Solution Steps

Combine -11x and -9x to get -20x.

$$5x^{2}-20x+7-14x^{2}$$

Combine 5x^{2} and -14x^{2} to get -9x^{2}.

$$-9x^{2}-20x+7$$

Factor

$$-9\left(x-\frac{-\sqrt{163}-10}{9}\right)\left(x-\frac{\sqrt{163}-10}{9}\right)$$

Steps Using the Quadratic Formula

Combine -11x and -9x to get -20x.

$$factor(5x^{2}-20x+7-14x^{2})$$

Combine 5x^{2} and -14x^{2} to get -9x^{2}.

$$factor(-9x^{2}-20x+7)$$

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.

$$-9x^{2}-20x+7=0$$

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.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-9\right)\times 7}}{2\left(-9\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-9\right)\times 7}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-20\right)±\sqrt{400+36\times 7}}{2\left(-9\right)}$$

Multiply 36 times 7.

$$x=\frac{-\left(-20\right)±\sqrt{400+252}}{2\left(-9\right)}$$

Add 400 to 252.

$$x=\frac{-\left(-20\right)±\sqrt{652}}{2\left(-9\right)}$$

Take the square root of 652.

$$x=\frac{-\left(-20\right)±2\sqrt{163}}{2\left(-9\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±2\sqrt{163}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{20±2\sqrt{163}}{-18}$$

Now solve the equation x=\frac{20±2\sqrt{163}}{-18} when ± is plus. Add 20 to 2\sqrt{163}.

$$x=\frac{2\sqrt{163}+20}{-18}$$

Divide 20+2\sqrt{163} by -18.

$$x=\frac{-\sqrt{163}-10}{9}$$

Now solve the equation x=\frac{20±2\sqrt{163}}{-18} when ± is minus. Subtract 2\sqrt{163} from 20.

$$x=\frac{20-2\sqrt{163}}{-18}$$

Divide 20-2\sqrt{163} by -18.

$$x=\frac{\sqrt{163}-10}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-10-\sqrt{163}}{9} for x_{1} and \frac{-10+\sqrt{163}}{9} for x_{2}.

$$-9x^{2}-20x+7=-9\left(x-\frac{-\sqrt{163}-10}{9}\right)\left(x-\frac{\sqrt{163}-10}{9}\right)$$