Solve x^2+7x-4

$${ x }^{ 2 } +7x-4$$

Factor

$$\left(x-\frac{-\sqrt{65}-7}{2}\right)\left(x-\frac{\sqrt{65}-7}{2}\right)$$

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.

$$x^{2}+7x-4=0$$

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.

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

Square 7.

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

Multiply -4 times -4.

$$x=\frac{-7±\sqrt{49+16}}{2}$$

Add 49 to 16.

$$x=\frac{-7±\sqrt{65}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is plus. Add -7 to \sqrt{65}.

$$x=\frac{\sqrt{65}-7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is minus. Subtract \sqrt{65} from -7.

$$x=\frac{-\sqrt{65}-7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+\sqrt{65}}{2} for x_{1} and \frac{-7-\sqrt{65}}{2} for x_{2}.

$$x^{2}+7x-4=\left(x-\frac{\sqrt{65}-7}{2}\right)\left(x-\frac{-\sqrt{65}-7}{2}\right)$$

Evaluate

$$x^{2}+7x-4$$