Solve 2x^2+10x+7x+35-3x^2-9x-2+6

$$2 { x }^{ 2 } +10x+7x+35-3 { x }^{ 2 } -9x-2+6$$

Evaluate

$$39+8x-x^{2}$$

Solution Steps

Combine 10x and 7x to get 17x.

$$2x^{2}+17x+35-3x^{2}-9x-2+6$$

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

$$-x^{2}+17x+35-9x-2+6$$

Combine 17x and -9x to get 8x.

$$-x^{2}+8x+35-2+6$$

Subtract 2 from 35 to get 33.

$$-x^{2}+8x+33+6$$

Add 33 and 6 to get 39.

$$-x^{2}+8x+39$$

Factor

$$-\left(x-\left(4-\sqrt{55}\right)\right)\left(x-\left(\sqrt{55}+4\right)\right)$$

Steps Using the Quadratic Formula

Combine 10x and 7x to get 17x.

$$factor(2x^{2}+17x+35-3x^{2}-9x-2+6)$$

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

$$factor(-x^{2}+17x+35-9x-2+6)$$

Combine 17x and -9x to get 8x.

$$factor(-x^{2}+8x+35-2+6)$$

Subtract 2 from 35 to get 33.

$$factor(-x^{2}+8x+33+6)$$

Add 33 and 6 to get 39.

$$factor(-x^{2}+8x+39)$$

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.

$$-x^{2}+8x+39=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\times 39}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\times 39}}{2\left(-1\right)}$$

Multiply 4 times 39.

$$x=\frac{-8±\sqrt{64+156}}{2\left(-1\right)}$$

Add 64 to 156.

$$x=\frac{-8±\sqrt{220}}{2\left(-1\right)}$$

Take the square root of 220.

$$x=\frac{-8±2\sqrt{55}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±2\sqrt{55}}{-2}$$

Now solve the equation x=\frac{-8±2\sqrt{55}}{-2} when ± is plus. Add -8 to 2\sqrt{55}.

$$x=\frac{2\sqrt{55}-8}{-2}$$

Divide -8+2\sqrt{55} by -2.

$$x=4-\sqrt{55}$$

Now solve the equation x=\frac{-8±2\sqrt{55}}{-2} when ± is minus. Subtract 2\sqrt{55} from -8.

$$x=\frac{-2\sqrt{55}-8}{-2}$$

Divide -8-2\sqrt{55} by -2.

$$x=\sqrt{55}+4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4-\sqrt{55} for x_{1} and 4+\sqrt{55} for x_{2}.

$$-x^{2}+8x+39=-\left(x-\left(4-\sqrt{55}\right)\right)\left(x-\left(\sqrt{55}+4\right)\right)$$