Solve 80y^2-17y^2-360y+4y+72y+405-19-81

$$80 { y }^{ 2 } -17 { y }^{ 2 } -360y+4y+72y+405-19-81$$

Evaluate

$$63y^{2}-284y+305$$

Solution Steps

Combine 80y^{2} and -17y^{2} to get 63y^{2}.

$$63y^{2}-360y+4y+72y+405-19-81$$

Combine -360y and 4y to get -356y.

$$63y^{2}-356y+72y+405-19-81$$

Combine -356y and 72y to get -284y.

$$63y^{2}-284y+405-19-81$$

Subtract 19 from 405 to get 386.

$$63y^{2}-284y+386-81$$

Subtract 81 from 386 to get 305.

$$63y^{2}-284y+305$$

Factor

$$63\left(y-\frac{142-\sqrt{949}}{63}\right)\left(y-\frac{\sqrt{949}+142}{63}\right)$$

Steps Using the Quadratic Formula

Combine 80y^{2} and -17y^{2} to get 63y^{2}.

$$factor(63y^{2}-360y+4y+72y+405-19-81)$$

Combine -360y and 4y to get -356y.

$$factor(63y^{2}-356y+72y+405-19-81)$$

Combine -356y and 72y to get -284y.

$$factor(63y^{2}-284y+405-19-81)$$

Subtract 19 from 405 to get 386.

$$factor(63y^{2}-284y+386-81)$$

Subtract 81 from 386 to get 305.

$$factor(63y^{2}-284y+305)$$

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.

$$63y^{2}-284y+305=0$$

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.

$$y=\frac{-\left(-284\right)±\sqrt{\left(-284\right)^{2}-4\times 63\times 305}}{2\times 63}$$

Square -284.

$$y=\frac{-\left(-284\right)±\sqrt{80656-4\times 63\times 305}}{2\times 63}$$

Multiply -4 times 63.

$$y=\frac{-\left(-284\right)±\sqrt{80656-252\times 305}}{2\times 63}$$

Multiply -252 times 305.

$$y=\frac{-\left(-284\right)±\sqrt{80656-76860}}{2\times 63}$$

Add 80656 to -76860.

$$y=\frac{-\left(-284\right)±\sqrt{3796}}{2\times 63}$$

Take the square root of 3796.

$$y=\frac{-\left(-284\right)±2\sqrt{949}}{2\times 63}$$

The opposite of -284 is 284.

$$y=\frac{284±2\sqrt{949}}{2\times 63}$$

Multiply 2 times 63.

$$y=\frac{284±2\sqrt{949}}{126}$$

Now solve the equation y=\frac{284±2\sqrt{949}}{126} when ± is plus. Add 284 to 2\sqrt{949}.

$$y=\frac{2\sqrt{949}+284}{126}$$

Divide 284+2\sqrt{949} by 126.

$$y=\frac{\sqrt{949}+142}{63}$$

Now solve the equation y=\frac{284±2\sqrt{949}}{126} when ± is minus. Subtract 2\sqrt{949} from 284.

$$y=\frac{284-2\sqrt{949}}{126}$$

Divide 284-2\sqrt{949} by 126.

$$y=\frac{142-\sqrt{949}}{63}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{142+\sqrt{949}}{63} for x_{1} and \frac{142-\sqrt{949}}{63} for x_{2}.

$$63y^{2}-284y+305=63\left(y-\frac{\sqrt{949}+142}{63}\right)\left(y-\frac{142-\sqrt{949}}{63}\right)$$