Solve 19+7x-4x^2

$$19 + 7 x - 4 x ^ { 2 }$$

Factor

$$-4\left(x-\frac{7-\sqrt{353}}{8}\right)\left(x-\frac{\sqrt{353}+7}{8}\right)$$

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.

$$-4x^{2}+7x+19=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-4\right)\times 19}}{2\left(-4\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-4\right)\times 19}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-7±\sqrt{49+16\times 19}}{2\left(-4\right)}$$

Multiply 16 times 19.

$$x=\frac{-7±\sqrt{49+304}}{2\left(-4\right)}$$

Add 49 to 304.

$$x=\frac{-7±\sqrt{353}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-7±\sqrt{353}}{-8}$$

Now solve the equation x=\frac{-7±\sqrt{353}}{-8} when ± is plus. Add -7 to \sqrt{353}.

$$x=\frac{\sqrt{353}-7}{-8}$$

Divide -7+\sqrt{353} by -8.

$$x=\frac{7-\sqrt{353}}{8}$$

Now solve the equation x=\frac{-7±\sqrt{353}}{-8} when ± is minus. Subtract \sqrt{353} from -7.

$$x=\frac{-\sqrt{353}-7}{-8}$$

Divide -7-\sqrt{353} by -8.

$$x=\frac{\sqrt{353}+7}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{353}}{8} for x_{1} and \frac{7+\sqrt{353}}{8} for x_{2}.

$$-4x^{2}+7x+19=-4\left(x-\frac{7-\sqrt{353}}{8}\right)\left(x-\frac{\sqrt{353}+7}{8}\right)$$

Evaluate

$$19+7x-4x^{2}$$