Solve 7x^2+8x-4+(-5x^2)+11

$$7 x ^ { 2 } + 8 x - 4 + ( - 5 x ^ { 2 } ) + 11$$

Evaluate

$$2x^{2}+8x+7$$

Solution Steps

Combine 7x^{2} and -5x^{2} to get 2x^{2}.

$$2x^{2}+8x-4+11$$

Add -4 and 11 to get 7.

$$2x^{2}+8x+7$$

Factor

$$2\left(x-\left(-\frac{\sqrt{2}}{2}-2\right)\right)\left(x-\left(\frac{\sqrt{2}}{2}-2\right)\right)$$

Steps Using the Quadratic Formula

Combine 7x^{2} and -5x^{2} to get 2x^{2}.

$$factor(2x^{2}+8x-4+11)$$

Add -4 and 11 to get 7.

$$factor(2x^{2}+8x+7)$$

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.

$$2x^{2}+8x+7=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 7}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 7}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\times 7}}{2\times 2}$$

Multiply -8 times 7.

$$x=\frac{-8±\sqrt{64-56}}{2\times 2}$$

Add 64 to -56.

$$x=\frac{-8±\sqrt{8}}{2\times 2}$$

Take the square root of 8.

$$x=\frac{-8±2\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±2\sqrt{2}}{4}$$

Now solve the equation x=\frac{-8±2\sqrt{2}}{4} when ± is plus. Add -8 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}-8}{4}$$

Divide 2\sqrt{2}-8 by 4.

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

Now solve the equation x=\frac{-8±2\sqrt{2}}{4} when ± is minus. Subtract 2\sqrt{2} from -8.

$$x=\frac{-2\sqrt{2}-8}{4}$$

Divide -8-2\sqrt{2} by 4.

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

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

$$2x^{2}+8x+7=2\left(x-\left(\frac{\sqrt{2}}{2}-2\right)\right)\left(x-\left(-\frac{\sqrt{2}}{2}-2\right)\right)$$