Solve {l}{432x^2+332x+56+221x^2-8x}{-16+96x^2+24x}

$$\left. \begin{array} { l } { 432 x ^ { 2 } + 332 x + 56 + 221 x ^ { 2 } - 8 x } \\ { - 16 + 96 x ^ { 2 } + 24 x } \end{array} \right.$$

Evaluate

$$749x^{2}+348x+40$$

Solution Steps

Combine 432x^{2} and 221x^{2} to get 653x^{2}.

$$653x^{2}+332x+56-8x-16+96x^{2}+24x$$

Combine 332x and -8x to get 324x.

$$653x^{2}+324x+56-16+96x^{2}+24x$$

Subtract 16 from 56 to get 40.

$$653x^{2}+324x+40+96x^{2}+24x$$

Combine 653x^{2} and 96x^{2} to get 749x^{2}.

$$749x^{2}+324x+40+24x$$

Combine 324x and 24x to get 348x.

$$749x^{2}+348x+40$$

Factor

$$749\left(x-\frac{-2\sqrt{79}-174}{749}\right)\left(x-\frac{2\sqrt{79}-174}{749}\right)$$

Steps Using the Quadratic Formula

Combine 432x^{2} and 221x^{2} to get 653x^{2}.

$$factor(653x^{2}+332x+56-8x-16+96x^{2}+24x)$$

Combine 332x and -8x to get 324x.

$$factor(653x^{2}+324x+56-16+96x^{2}+24x)$$

Subtract 16 from 56 to get 40.

$$factor(653x^{2}+324x+40+96x^{2}+24x)$$

Combine 653x^{2} and 96x^{2} to get 749x^{2}.

$$factor(749x^{2}+324x+40+24x)$$

Combine 324x and 24x to get 348x.

$$factor(749x^{2}+348x+40)$$

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.

$$749x^{2}+348x+40=0$$

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.

$$x=\frac{-348±\sqrt{348^{2}-4\times 749\times 40}}{2\times 749}$$

Square 348.

$$x=\frac{-348±\sqrt{121104-4\times 749\times 40}}{2\times 749}$$

Multiply -4 times 749.

$$x=\frac{-348±\sqrt{121104-2996\times 40}}{2\times 749}$$

Multiply -2996 times 40.

$$x=\frac{-348±\sqrt{121104-119840}}{2\times 749}$$

Add 121104 to -119840.

$$x=\frac{-348±\sqrt{1264}}{2\times 749}$$

Take the square root of 1264.

$$x=\frac{-348±4\sqrt{79}}{2\times 749}$$

Multiply 2 times 749.

$$x=\frac{-348±4\sqrt{79}}{1498}$$

Now solve the equation x=\frac{-348±4\sqrt{79}}{1498} when ± is plus. Add -348 to 4\sqrt{79}.

$$x=\frac{4\sqrt{79}-348}{1498}$$

Divide -348+4\sqrt{79} by 1498.

$$x=\frac{2\sqrt{79}-174}{749}$$

Now solve the equation x=\frac{-348±4\sqrt{79}}{1498} when ± is minus. Subtract 4\sqrt{79} from -348.

$$x=\frac{-4\sqrt{79}-348}{1498}$$

Divide -348-4\sqrt{79} by 1498.

$$x=\frac{-2\sqrt{79}-174}{749}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-174+2\sqrt{79}}{749} for x_{1} and \frac{-174-2\sqrt{79}}{749} for x_{2}.

$$749x^{2}+348x+40=749\left(x-\frac{2\sqrt{79}-174}{749}\right)\left(x-\frac{-2\sqrt{79}-174}{749}\right)$$