Solve 12x+4-x^2-9

$$12 x + 4 - x ^ { 2 } - 9$$

Factor

$$-\left(x-\left(6-\sqrt{31}\right)\right)\left(x-\left(\sqrt{31}+6\right)\right)$$

Steps Using the Quadratic Formula

Subtract 9 from 4 to get -5.

$$factor(12x-5-x^{2})$$

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.

$$-x^{2}+12x-5=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$x=\frac{-12±\sqrt{144-20}}{2\left(-1\right)}$$

Add 144 to -20.

$$x=\frac{-12±\sqrt{124}}{2\left(-1\right)}$$

Take the square root of 124.

$$x=\frac{-12±2\sqrt{31}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±2\sqrt{31}}{-2}$$

Now solve the equation x=\frac{-12±2\sqrt{31}}{-2} when ± is plus. Add -12 to 2\sqrt{31}.

$$x=\frac{2\sqrt{31}-12}{-2}$$

Divide -12+2\sqrt{31} by -2.

$$x=6-\sqrt{31}$$

Now solve the equation x=\frac{-12±2\sqrt{31}}{-2} when ± is minus. Subtract 2\sqrt{31} from -12.

$$x=\frac{-2\sqrt{31}-12}{-2}$$

Divide -12-2\sqrt{31} by -2.

$$x=\sqrt{31}+6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6-\sqrt{31} for x_{1} and 6+\sqrt{31} for x_{2}.

$$-x^{2}+12x-5=-\left(x-\left(6-\sqrt{31}\right)\right)\left(x-\left(\sqrt{31}+6\right)\right)$$

Evaluate

$$-x^{2}+12x-5$$

Solution Steps

Subtract 9 from 4 to get -5.

$$12x-5-x^{2}$$