Solve f(x)=6x-12x^2-5x+4

$$f ( x ) = 6 x - 12 x ^ { 2 } - 5 x + 4$$

Factor

$$-12\left(x-\frac{1-\sqrt{193}}{24}\right)\left(x-\frac{\sqrt{193}+1}{24}\right)$$

Steps Using the Quadratic Formula

Combine 6x and -5x to get x.

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

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.

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

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.

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

Square 1.

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

Multiply -4 times -12.

$$x=\frac{-1±\sqrt{1+48\times 4}}{2\left(-12\right)}$$

Multiply 48 times 4.

$$x=\frac{-1±\sqrt{1+192}}{2\left(-12\right)}$$

Add 1 to 192.

$$x=\frac{-1±\sqrt{193}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-1±\sqrt{193}}{-24}$$

Now solve the equation x=\frac{-1±\sqrt{193}}{-24} when ± is plus. Add -1 to \sqrt{193}.

$$x=\frac{\sqrt{193}-1}{-24}$$

Divide -1+\sqrt{193} by -24.

$$x=\frac{1-\sqrt{193}}{24}$$

Now solve the equation x=\frac{-1±\sqrt{193}}{-24} when ± is minus. Subtract \sqrt{193} from -1.

$$x=\frac{-\sqrt{193}-1}{-24}$$

Divide -1-\sqrt{193} by -24.

$$x=\frac{\sqrt{193}+1}{24}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{193}}{24} for x_{1} and \frac{1+\sqrt{193}}{24} for x_{2}.

$$-12x^{2}+x+4=-12\left(x-\frac{1-\sqrt{193}}{24}\right)\left(x-\frac{\sqrt{193}+1}{24}\right)$$

Evaluate

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

Solution Steps

Combine 6x and -5x to get x.

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