Solve x^2+15x^2-27x-20

$$x ^ { 2 } + 15 x ^ { 2 } - 27 x - 20$$

Factor

$$16\left(x-\frac{27-7\sqrt{41}}{32}\right)\left(x-\frac{7\sqrt{41}+27}{32}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and 15x^{2} to get 16x^{2}.

$$factor(16x^{2}-27x-20)$$

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.

$$16x^{2}-27x-20=0$$

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.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 16\left(-20\right)}}{2\times 16}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\times 16\left(-20\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-27\right)±\sqrt{729-64\left(-20\right)}}{2\times 16}$$

Multiply -64 times -20.

$$x=\frac{-\left(-27\right)±\sqrt{729+1280}}{2\times 16}$$

Add 729 to 1280.

$$x=\frac{-\left(-27\right)±\sqrt{2009}}{2\times 16}$$

Take the square root of 2009.

$$x=\frac{-\left(-27\right)±7\sqrt{41}}{2\times 16}$$

The opposite of -27 is 27.

$$x=\frac{27±7\sqrt{41}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{27±7\sqrt{41}}{32}$$

Now solve the equation x=\frac{27±7\sqrt{41}}{32} when ± is plus. Add 27 to 7\sqrt{41}.

$$x=\frac{7\sqrt{41}+27}{32}$$

Now solve the equation x=\frac{27±7\sqrt{41}}{32} when ± is minus. Subtract 7\sqrt{41} from 27.

$$x=\frac{27-7\sqrt{41}}{32}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{27+7\sqrt{41}}{32} for x_{1} and \frac{27-7\sqrt{41}}{32} for x_{2}.

$$16x^{2}-27x-20=16\left(x-\frac{7\sqrt{41}+27}{32}\right)\left(x-\frac{27-7\sqrt{41}}{32}\right)$$

Evaluate

$$16x^{2}-27x-20$$

Solution Steps

Combine x^{2} and 15x^{2} to get 16x^{2}.

$$16x^{2}-27x-20$$