Solve -4x^2+12x+1

$$-4 { x }^{ 2 } +12x+1$$

Factor

$$-4\left(x-\frac{3-\sqrt{10}}{2}\right)\left(x-\frac{\sqrt{10}+3}{2}\right)$$

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.

$$-4x^{2}+12x+1=0$$

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.

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

Square 12.

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

Multiply -4 times -4.

$$x=\frac{-12±\sqrt{144+16}}{2\left(-4\right)}$$

Add 144 to 16.

$$x=\frac{-12±\sqrt{160}}{2\left(-4\right)}$$

Take the square root of 160.

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

Multiply 2 times -4.

$$x=\frac{-12±4\sqrt{10}}{-8}$$

Now solve the equation x=\frac{-12±4\sqrt{10}}{-8} when ± is plus. Add -12 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}-12}{-8}$$

Divide -12+4\sqrt{10} by -8.

$$x=\frac{3-\sqrt{10}}{2}$$

Now solve the equation x=\frac{-12±4\sqrt{10}}{-8} when ± is minus. Subtract 4\sqrt{10} from -12.

$$x=\frac{-4\sqrt{10}-12}{-8}$$

Divide -12-4\sqrt{10} by -8.

$$x=\frac{\sqrt{10}+3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{10}}{2} for x_{1} and \frac{3+\sqrt{10}}{2} for x_{2}.

$$-4x^{2}+12x+1=-4\left(x-\frac{3-\sqrt{10}}{2}\right)\left(x-\frac{\sqrt{10}+3}{2}\right)$$

Evaluate

$$1+12x-4x^{2}$$