Solve 9x^2-6-10x^2+10+12x

$$9 x ^ { 2 } - 6 - 10 x ^ { 2 } + 10 + 12 x$$

Evaluate

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

Solution Steps

Combine 9x^{2} and -10x^{2} to get -x^{2}.

$$-x^{2}-6+10+12x$$

Add -6 and 10 to get 4.

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

Factor

$$-\left(x-\left(6-2\sqrt{10}\right)\right)\left(x-\left(2\sqrt{10}+6\right)\right)$$

Steps Using the Quadratic Formula

Combine 9x^{2} and -10x^{2} to get -x^{2}.

$$factor(-x^{2}-6+10+12x)$$

Add -6 and 10 to get 4.

$$factor(-x^{2}+4+12x)$$

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.

$$-x^{2}+12x+4=0$$

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.

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

Square 12.

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

Multiply -4 times -1.

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

Multiply 4 times 4.

$$x=\frac{-12±\sqrt{144+16}}{2\left(-1\right)}$$

Add 144 to 16.

$$x=\frac{-12±\sqrt{160}}{2\left(-1\right)}$$

Take the square root of 160.

$$x=\frac{-12±4\sqrt{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±4\sqrt{10}}{-2}$$

Now solve the equation x=\frac{-12±4\sqrt{10}}{-2} when ± is plus. Add -12 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}-12}{-2}$$

Divide -12+4\sqrt{10} by -2.

$$x=6-2\sqrt{10}$$

Now solve the equation x=\frac{-12±4\sqrt{10}}{-2} when ± is minus. Subtract 4\sqrt{10} from -12.

$$x=\frac{-4\sqrt{10}-12}{-2}$$

Divide -12-4\sqrt{10} by -2.

$$x=2\sqrt{10}+6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6-2\sqrt{10} for x_{1} and 6+2\sqrt{10} for x_{2}.

$$-x^{2}+12x+4=-\left(x-\left(6-2\sqrt{10}\right)\right)\left(x-\left(2\sqrt{10}+6\right)\right)$$