Solve 7a+4-a^2

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

Factor

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

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.

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

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.

$$a=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Square 7.

$$a=\frac{-7±\sqrt{49-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-7±\sqrt{49+4\times 4}}{2\left(-1\right)}$$

Multiply 4 times 4.

$$a=\frac{-7±\sqrt{49+16}}{2\left(-1\right)}$$

Add 49 to 16.

$$a=\frac{-7±\sqrt{65}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-7±\sqrt{65}}{-2}$$

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

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

Divide -7+\sqrt{65} by -2.

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

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

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

Divide -7-\sqrt{65} by -2.

$$a=\frac{\sqrt{65}+7}{2}$$

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}.

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

Evaluate

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