Solve -6z^2-z^2+3z+1+z^2-z

$$- 6 z ^ { 2 } - z ^ { 2 } + 3 z + 1 + z ^ { 2 } - z$$

Evaluate

$$1+2z-6z^{2}$$

Solution Steps

Combine -6z^{2} and -z^{2} to get -7z^{2}.

$$-7z^{2}+3z+1+z^{2}-z$$

Combine -7z^{2} and z^{2} to get -6z^{2}.

$$-6z^{2}+3z+1-z$$

Combine 3z and -z to get 2z.

$$-6z^{2}+2z+1$$

Factor

$$-6\left(z-\frac{1-\sqrt{7}}{6}\right)\left(z-\frac{\sqrt{7}+1}{6}\right)$$

Steps Using the Quadratic Formula

Combine -6z^{2} and -z^{2} to get -7z^{2}.

$$factor(-7z^{2}+3z+1+z^{2}-z)$$

Combine -7z^{2} and z^{2} to get -6z^{2}.

$$factor(-6z^{2}+3z+1-z)$$

Combine 3z and -z to get 2z.

$$factor(-6z^{2}+2z+1)$$

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.

$$-6z^{2}+2z+1=0$$

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.

$$z=\frac{-2±\sqrt{2^{2}-4\left(-6\right)}}{2\left(-6\right)}$$

Square 2.

$$z=\frac{-2±\sqrt{4-4\left(-6\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$z=\frac{-2±\sqrt{4+24}}{2\left(-6\right)}$$

Add 4 to 24.

$$z=\frac{-2±\sqrt{28}}{2\left(-6\right)}$$

Take the square root of 28.

$$z=\frac{-2±2\sqrt{7}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$z=\frac{-2±2\sqrt{7}}{-12}$$

Now solve the equation z=\frac{-2±2\sqrt{7}}{-12} when ± is plus. Add -2 to 2\sqrt{7}.

$$z=\frac{2\sqrt{7}-2}{-12}$$

Divide -2+2\sqrt{7} by -12.

$$z=\frac{1-\sqrt{7}}{6}$$

Now solve the equation z=\frac{-2±2\sqrt{7}}{-12} when ± is minus. Subtract 2\sqrt{7} from -2.

$$z=\frac{-2\sqrt{7}-2}{-12}$$

Divide -2-2\sqrt{7} by -12.

$$z=\frac{\sqrt{7}+1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{7}}{6} for x_{1} and \frac{1+\sqrt{7}}{6} for x_{2}.

$$-6z^{2}+2z+1=-6\left(z-\frac{1-\sqrt{7}}{6}\right)\left(z-\frac{\sqrt{7}+1}{6}\right)$$