Solve 19p+4-9-11p^2

$$19 p + 4 - 9 - 11 p ^ { 2 }$$

Factor

$$-11\left(p-\frac{19-\sqrt{141}}{22}\right)\left(p-\frac{\sqrt{141}+19}{22}\right)$$

Steps Using the Quadratic Formula

Subtract 9 from 4 to get -5.

$$factor(19p-5-11p^{2})$$

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.

$$-11p^{2}+19p-5=0$$

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.

$$p=\frac{-19±\sqrt{19^{2}-4\left(-11\right)\left(-5\right)}}{2\left(-11\right)}$$

Square 19.

$$p=\frac{-19±\sqrt{361-4\left(-11\right)\left(-5\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$p=\frac{-19±\sqrt{361+44\left(-5\right)}}{2\left(-11\right)}$$

Multiply 44 times -5.

$$p=\frac{-19±\sqrt{361-220}}{2\left(-11\right)}$$

Add 361 to -220.

$$p=\frac{-19±\sqrt{141}}{2\left(-11\right)}$$

Multiply 2 times -11.

$$p=\frac{-19±\sqrt{141}}{-22}$$

Now solve the equation p=\frac{-19±\sqrt{141}}{-22} when ± is plus. Add -19 to \sqrt{141}.

$$p=\frac{\sqrt{141}-19}{-22}$$

Divide -19+\sqrt{141} by -22.

$$p=\frac{19-\sqrt{141}}{22}$$

Now solve the equation p=\frac{-19±\sqrt{141}}{-22} when ± is minus. Subtract \sqrt{141} from -19.

$$p=\frac{-\sqrt{141}-19}{-22}$$

Divide -19-\sqrt{141} by -22.

$$p=\frac{\sqrt{141}+19}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19-\sqrt{141}}{22} for x_{1} and \frac{19+\sqrt{141}}{22} for x_{2}.

$$-11p^{2}+19p-5=-11\left(p-\frac{19-\sqrt{141}}{22}\right)\left(p-\frac{\sqrt{141}+19}{22}\right)$$

Evaluate

$$-11p^{2}+19p-5$$

Solution Steps

Subtract 9 from 4 to get -5.

$$19p-5-11p^{2}$$