Solve 1-8y+y^2-68+4y-71-7y

$$1 - 8 y + y ^ { 2 } - 68 + 4 y - 71 - 7 y$$

Evaluate

$$y^{2}-11y-138$$

Solution Steps

Subtract 68 from 1 to get -67.

$$-67-8y+y^{2}+4y-71-7y$$

Combine -8y and 4y to get -4y.

$$-67-4y+y^{2}-71-7y$$

Subtract 71 from -67 to get -138.

$$-138-4y+y^{2}-7y$$

Combine -4y and -7y to get -11y.

$$-138-11y+y^{2}$$

Factor

$$\left(y-\frac{11-\sqrt{673}}{2}\right)\left(y-\frac{\sqrt{673}+11}{2}\right)$$

Steps Using the Quadratic Formula

Subtract 68 from 1 to get -67.

$$factor(-67-8y+y^{2}+4y-71-7y)$$

Combine -8y and 4y to get -4y.

$$factor(-67-4y+y^{2}-71-7y)$$

Subtract 71 from -67 to get -138.

$$factor(-138-4y+y^{2}-7y)$$

Combine -4y and -7y to get -11y.

$$factor(-138-11y+y^{2})$$

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.

$$y^{2}-11y-138=0$$

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.

$$y=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-138\right)}}{2}$$

Square -11.

$$y=\frac{-\left(-11\right)±\sqrt{121-4\left(-138\right)}}{2}$$

Multiply -4 times -138.

$$y=\frac{-\left(-11\right)±\sqrt{121+552}}{2}$$

Add 121 to 552.

$$y=\frac{-\left(-11\right)±\sqrt{673}}{2}$$

The opposite of -11 is 11.

$$y=\frac{11±\sqrt{673}}{2}$$

Now solve the equation y=\frac{11±\sqrt{673}}{2} when ± is plus. Add 11 to \sqrt{673}.

$$y=\frac{\sqrt{673}+11}{2}$$

Now solve the equation y=\frac{11±\sqrt{673}}{2} when ± is minus. Subtract \sqrt{673} from 11.

$$y=\frac{11-\sqrt{673}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{673}}{2} for x_{1} and \frac{11-\sqrt{673}}{2} for x_{2}.

$$y^{2}-11y-138=\left(y-\frac{\sqrt{673}+11}{2}\right)\left(y-\frac{11-\sqrt{673}}{2}\right)$$