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

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

Evaluate

$$11+12x-2x^{2}$$

Solution Steps

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

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

Combine 4x and 8x to get 12x.

$$-2x^{2}+12x+11$$

Factor

$$-2\left(x-\left(-\frac{\sqrt{58}}{2}+3\right)\right)\left(x-\left(\frac{\sqrt{58}}{2}+3\right)\right)$$

Steps Using the Quadratic Formula

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

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

Combine 4x and 8x to get 12x.

$$factor(-2x^{2}+12x+11)$$

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.

$$-2x^{2}+12x+11=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-12±\sqrt{144+8\times 11}}{2\left(-2\right)}$$

Multiply 8 times 11.

$$x=\frac{-12±\sqrt{144+88}}{2\left(-2\right)}$$

Add 144 to 88.

$$x=\frac{-12±\sqrt{232}}{2\left(-2\right)}$$

Take the square root of 232.

$$x=\frac{-12±2\sqrt{58}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-12±2\sqrt{58}}{-4}$$

Now solve the equation x=\frac{-12±2\sqrt{58}}{-4} when ± is plus. Add -12 to 2\sqrt{58}.

$$x=\frac{2\sqrt{58}-12}{-4}$$

Divide -12+2\sqrt{58} by -4.

$$x=-\frac{\sqrt{58}}{2}+3$$

Now solve the equation x=\frac{-12±2\sqrt{58}}{-4} when ± is minus. Subtract 2\sqrt{58} from -12.

$$x=\frac{-2\sqrt{58}-12}{-4}$$

Divide -12-2\sqrt{58} by -4.

$$x=\frac{\sqrt{58}}{2}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\frac{\sqrt{58}}{2} for x_{1} and 3+\frac{\sqrt{58}}{2} for x_{2}.

$$-2x^{2}+12x+11=-2\left(x-\left(-\frac{\sqrt{58}}{2}+3\right)\right)\left(x-\left(\frac{\sqrt{58}}{2}+3\right)\right)$$