Solve 4x^2-9-15x^2+11+11x

$$4 x ^ { 2 } - 9 - 15 x ^ { 2 } + 11 + 11 x$$

Evaluate

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

Solution Steps

Combine 4x^{2} and -15x^{2} to get -11x^{2}.

$$-11x^{2}-9+11+11x$$

Add -9 and 11 to get 2.

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

Factor

$$-11\left(x-\left(-\frac{\sqrt{209}}{22}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{209}}{22}+\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and -15x^{2} to get -11x^{2}.

$$factor(-11x^{2}-9+11+11x)$$

Add -9 and 11 to get 2.

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

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.

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

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.

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

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-11\right)\times 2}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-11±\sqrt{121+44\times 2}}{2\left(-11\right)}$$

Multiply 44 times 2.

$$x=\frac{-11±\sqrt{121+88}}{2\left(-11\right)}$$

Add 121 to 88.

$$x=\frac{-11±\sqrt{209}}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{-11±\sqrt{209}}{-22}$$

Now solve the equation x=\frac{-11±\sqrt{209}}{-22} when ± is plus. Add -11 to \sqrt{209}.

$$x=\frac{\sqrt{209}-11}{-22}$$

Divide -11+\sqrt{209} by -22.

$$x=-\frac{\sqrt{209}}{22}+\frac{1}{2}$$

Now solve the equation x=\frac{-11±\sqrt{209}}{-22} when ± is minus. Subtract \sqrt{209} from -11.

$$x=\frac{-\sqrt{209}-11}{-22}$$

Divide -11-\sqrt{209} by -22.

$$x=\frac{\sqrt{209}}{22}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}-\frac{\sqrt{209}}{22} for x_{1} and \frac{1}{2}+\frac{\sqrt{209}}{22} for x_{2}.

$$-11x^{2}+11x+2=-11\left(x-\left(-\frac{\sqrt{209}}{22}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{209}}{22}+\frac{1}{2}\right)\right)$$