Solve =35+8x-3x^2-4x^2+12x-9

$$= 35 + 8 x - 3 x ^ { 2 } - 4 x ^ { 2 } + 12 x - 9$$

Evaluate

$$26+20x-7x^{2}$$

Solution Steps

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$35+8x-7x^{2}+12x-9$$

Combine 8x and 12x to get 20x.

$$35+20x-7x^{2}-9$$

Subtract 9 from 35 to get 26.

$$26+20x-7x^{2}$$

Factor

$$-7\left(x-\frac{10-\sqrt{282}}{7}\right)\left(x-\frac{\sqrt{282}+10}{7}\right)$$

Steps Using the Quadratic Formula

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$factor(35+8x-7x^{2}+12x-9)$$

Combine 8x and 12x to get 20x.

$$factor(35+20x-7x^{2}-9)$$

Subtract 9 from 35 to get 26.

$$factor(26+20x-7x^{2})$$

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.

$$-7x^{2}+20x+26=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-7\right)\times 26}}{2\left(-7\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-7\right)\times 26}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-20±\sqrt{400+28\times 26}}{2\left(-7\right)}$$

Multiply 28 times 26.

$$x=\frac{-20±\sqrt{400+728}}{2\left(-7\right)}$$

Add 400 to 728.

$$x=\frac{-20±\sqrt{1128}}{2\left(-7\right)}$$

Take the square root of 1128.

$$x=\frac{-20±2\sqrt{282}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-20±2\sqrt{282}}{-14}$$

Now solve the equation x=\frac{-20±2\sqrt{282}}{-14} when ± is plus. Add -20 to 2\sqrt{282}.

$$x=\frac{2\sqrt{282}-20}{-14}$$

Divide -20+2\sqrt{282} by -14.

$$x=\frac{10-\sqrt{282}}{7}$$

Now solve the equation x=\frac{-20±2\sqrt{282}}{-14} when ± is minus. Subtract 2\sqrt{282} from -20.

$$x=\frac{-2\sqrt{282}-20}{-14}$$

Divide -20-2\sqrt{282} by -14.

$$x=\frac{\sqrt{282}+10}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{10-\sqrt{282}}{7} for x_{1} and \frac{10+\sqrt{282}}{7} for x_{2}.

$$-7x^{2}+20x+26=-7\left(x-\frac{10-\sqrt{282}}{7}\right)\left(x-\frac{\sqrt{282}+10}{7}\right)$$