Solve 3-7x-4x^2

$$3 - 7 x - 4 x ^ { 2 }$$

Factor

$$-4\left(x-\frac{-\sqrt{97}-7}{8}\right)\left(x-\frac{\sqrt{97}-7}{8}\right)$$

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.

$$-4x^{2}-7x+3=0$$

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.

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

Square -7.

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

Multiply -4 times -4.

$$x=\frac{-\left(-7\right)±\sqrt{49+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49+48}}{2\left(-4\right)}$$

Add 49 to 48.

$$x=\frac{-\left(-7\right)±\sqrt{97}}{2\left(-4\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{97}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{7±\sqrt{97}}{-8}$$

Now solve the equation x=\frac{7±\sqrt{97}}{-8} when ± is plus. Add 7 to \sqrt{97}.

$$x=\frac{\sqrt{97}+7}{-8}$$

Divide 7+\sqrt{97} by -8.

$$x=\frac{-\sqrt{97}-7}{8}$$

Now solve the equation x=\frac{7±\sqrt{97}}{-8} when ± is minus. Subtract \sqrt{97} from 7.

$$x=\frac{7-\sqrt{97}}{-8}$$

Divide 7-\sqrt{97} by -8.

$$x=\frac{\sqrt{97}-7}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7-\sqrt{97}}{8} for x_{1} and \frac{-7+\sqrt{97}}{8} for x_{2}.

$$-4x^{2}-7x+3=-4\left(x-\frac{-\sqrt{97}-7}{8}\right)\left(x-\frac{\sqrt{97}-7}{8}\right)$$

Evaluate

$$3-7x-4x^{2}$$