Solve 3x^2+224x-3136

$$3 { x }^{ 2 } +224x-3136$$

Factor

$$3\left(x-\frac{-56\sqrt{7}-112}{3}\right)\left(x-\frac{56\sqrt{7}-112}{3}\right)$$

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.

$$3x^{2}+224x-3136=0$$

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.

$$x=\frac{-224±\sqrt{224^{2}-4\times 3\left(-3136\right)}}{2\times 3}$$

Square 224.

$$x=\frac{-224±\sqrt{50176-4\times 3\left(-3136\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-224±\sqrt{50176-12\left(-3136\right)}}{2\times 3}$$

Multiply -12 times -3136.

$$x=\frac{-224±\sqrt{50176+37632}}{2\times 3}$$

Add 50176 to 37632.

$$x=\frac{-224±\sqrt{87808}}{2\times 3}$$

Take the square root of 87808.

$$x=\frac{-224±112\sqrt{7}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-224±112\sqrt{7}}{6}$$

Now solve the equation x=\frac{-224±112\sqrt{7}}{6} when ± is plus. Add -224 to 112\sqrt{7}.

$$x=\frac{112\sqrt{7}-224}{6}$$

Divide -224+112\sqrt{7} by 6.

$$x=\frac{56\sqrt{7}-112}{3}$$

Now solve the equation x=\frac{-224±112\sqrt{7}}{6} when ± is minus. Subtract 112\sqrt{7} from -224.

$$x=\frac{-112\sqrt{7}-224}{6}$$

Divide -224-112\sqrt{7} by 6.

$$x=\frac{-56\sqrt{7}-112}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-112+56\sqrt{7}}{3} for x_{1} and \frac{-112-56\sqrt{7}}{3} for x_{2}.

$$3x^{2}+224x-3136=3\left(x-\frac{56\sqrt{7}-112}{3}\right)\left(x-\frac{-56\sqrt{7}-112}{3}\right)$$

Evaluate

$$3x^{2}+224x-3136$$