Solve 3x^2-2-17x^2+13+10x

$$3 x ^ { 2 } - 2 - 17 x ^ { 2 } + 13 + 10 x$$

Evaluate

$$11+10x-14x^{2}$$

Solution Steps

Combine 3x^{2} and -17x^{2} to get -14x^{2}.

$$-14x^{2}-2+13+10x$$

Add -2 and 13 to get 11.

$$-14x^{2}+11+10x$$

Factor

$$-14\left(x-\frac{5-\sqrt{179}}{14}\right)\left(x-\frac{\sqrt{179}+5}{14}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and -17x^{2} to get -14x^{2}.

$$factor(-14x^{2}-2+13+10x)$$

Add -2 and 13 to get 11.

$$factor(-14x^{2}+11+10x)$$

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.

$$-14x^{2}+10x+11=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-14\right)\times 11}}{2\left(-14\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-14\right)\times 11}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-10±\sqrt{100+56\times 11}}{2\left(-14\right)}$$

Multiply 56 times 11.

$$x=\frac{-10±\sqrt{100+616}}{2\left(-14\right)}$$

Add 100 to 616.

$$x=\frac{-10±\sqrt{716}}{2\left(-14\right)}$$

Take the square root of 716.

$$x=\frac{-10±2\sqrt{179}}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-10±2\sqrt{179}}{-28}$$

Now solve the equation x=\frac{-10±2\sqrt{179}}{-28} when ± is plus. Add -10 to 2\sqrt{179}.

$$x=\frac{2\sqrt{179}-10}{-28}$$

Divide -10+2\sqrt{179} by -28.

$$x=\frac{5-\sqrt{179}}{14}$$

Now solve the equation x=\frac{-10±2\sqrt{179}}{-28} when ± is minus. Subtract 2\sqrt{179} from -10.

$$x=\frac{-2\sqrt{179}-10}{-28}$$

Divide -10-2\sqrt{179} by -28.

$$x=\frac{\sqrt{179}+5}{14}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{179}}{14} for x_{1} and \frac{5+\sqrt{179}}{14} for x_{2}.

$$-14x^{2}+10x+11=-14\left(x-\frac{5-\sqrt{179}}{14}\right)\left(x-\frac{\sqrt{179}+5}{14}\right)$$