Solve 3x^2+7x-21

$$3 { x }^{ 2 } +7x-21$$

Factor

$$3\left(x-\frac{-\sqrt{301}-7}{6}\right)\left(x-\frac{\sqrt{301}-7}{6}\right)$$

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.

$$3x^{2}+7x-21=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\left(-21\right)}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3\left(-21\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12\left(-21\right)}}{2\times 3}$$

Multiply -12 times -21.

$$x=\frac{-7±\sqrt{49+252}}{2\times 3}$$

Add 49 to 252.

$$x=\frac{-7±\sqrt{301}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±\sqrt{301}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{301}}{6} when ± is plus. Add -7 to \sqrt{301}.

$$x=\frac{\sqrt{301}-7}{6}$$

Now solve the equation x=\frac{-7±\sqrt{301}}{6} when ± is minus. Subtract \sqrt{301} from -7.

$$x=\frac{-\sqrt{301}-7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{301}}{6} for x_{1} and \frac{-7-\sqrt{301}}{6} for x_{2}.

$$3x^{2}+7x-21=3\left(x-\frac{\sqrt{301}-7}{6}\right)\left(x-\frac{-\sqrt{301}-7}{6}\right)$$

Evaluate

$$3x^{2}+7x-21$$