Solve 6^4x+23x^2+15=

$${ 6 }^{ 4 } x+23 { x }^{ 2 } +15=$$

Factor

$$23\left(x-\frac{-\sqrt{419559}-648}{23}\right)\left(x-\frac{\sqrt{419559}-648}{23}\right)$$

Steps Using the Quadratic Formula

Calculate 6 to the power of 4 and get 1296.

$$factor(1296x+23x^{2}+15)$$

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.

$$23x^{2}+1296x+15=0$$

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.

$$x=\frac{-1296±\sqrt{1296^{2}-4\times 23\times 15}}{2\times 23}$$

Square 1296.

$$x=\frac{-1296±\sqrt{1679616-4\times 23\times 15}}{2\times 23}$$

Multiply -4 times 23.

$$x=\frac{-1296±\sqrt{1679616-92\times 15}}{2\times 23}$$

Multiply -92 times 15.

$$x=\frac{-1296±\sqrt{1679616-1380}}{2\times 23}$$

Add 1679616 to -1380.

$$x=\frac{-1296±\sqrt{1678236}}{2\times 23}$$

Take the square root of 1678236.

$$x=\frac{-1296±2\sqrt{419559}}{2\times 23}$$

Multiply 2 times 23.

$$x=\frac{-1296±2\sqrt{419559}}{46}$$

Now solve the equation x=\frac{-1296±2\sqrt{419559}}{46} when ± is plus. Add -1296 to 2\sqrt{419559}.

$$x=\frac{2\sqrt{419559}-1296}{46}$$

Divide -1296+2\sqrt{419559} by 46.

$$x=\frac{\sqrt{419559}-648}{23}$$

Now solve the equation x=\frac{-1296±2\sqrt{419559}}{46} when ± is minus. Subtract 2\sqrt{419559} from -1296.

$$x=\frac{-2\sqrt{419559}-1296}{46}$$

Divide -1296-2\sqrt{419559} by 46.

$$x=\frac{-\sqrt{419559}-648}{23}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-648+\sqrt{419559}}{23} for x_{1} and \frac{-648-\sqrt{419559}}{23} for x_{2}.

$$23x^{2}+1296x+15=23\left(x-\frac{\sqrt{419559}-648}{23}\right)\left(x-\frac{-\sqrt{419559}-648}{23}\right)$$

Evaluate

$$23x^{2}+1296x+15$$

Solution Steps

Calculate 6 to the power of 4 and get 1296.

$$1296x+23x^{2}+15$$