Solve 24x-14+7x^2

$$24x-14+7 { x }^{ 2 }$$

Factor

$$7\left(x-\frac{-11\sqrt{2}-12}{7}\right)\left(x-\frac{11\sqrt{2}-12}{7}\right)$$

Steps Using the Quadratic Formula

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.

$$7x^{2}+24x-14=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 7\left(-14\right)}}{2\times 7}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 7\left(-14\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-24±\sqrt{576-28\left(-14\right)}}{2\times 7}$$

Multiply -28 times -14.

$$x=\frac{-24±\sqrt{576+392}}{2\times 7}$$

Add 576 to 392.

$$x=\frac{-24±\sqrt{968}}{2\times 7}$$

Take the square root of 968.

$$x=\frac{-24±22\sqrt{2}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-24±22\sqrt{2}}{14}$$

Now solve the equation x=\frac{-24±22\sqrt{2}}{14} when ± is plus. Add -24 to 22\sqrt{2}.

$$x=\frac{22\sqrt{2}-24}{14}$$

Divide -24+22\sqrt{2} by 14.

$$x=\frac{11\sqrt{2}-12}{7}$$

Now solve the equation x=\frac{-24±22\sqrt{2}}{14} when ± is minus. Subtract 22\sqrt{2} from -24.

$$x=\frac{-22\sqrt{2}-24}{14}$$

Divide -24-22\sqrt{2} by 14.

$$x=\frac{-11\sqrt{2}-12}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-12+11\sqrt{2}}{7} for x_{1} and \frac{-12-11\sqrt{2}}{7} for x_{2}.

$$7x^{2}+24x-14=7\left(x-\frac{11\sqrt{2}-12}{7}\right)\left(x-\frac{-11\sqrt{2}-12}{7}\right)$$

Evaluate

$$7x^{2}+24x-14$$