Solve -72x^2+17x-1

$$-72 { x }^{ 2 } +17x-1$$

Factor

$$\left(1-9x\right)\left(8x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -72x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=17 ab=-72\left(-1\right)=72$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 17.

$$a=9 b=8$$

Rewrite -72x^{2}+17x-1 as \left(-72x^{2}+9x\right)+\left(8x-1\right).

$$\left(-72x^{2}+9x\right)+\left(8x-1\right)$$

Factor out -9x in -72x^{2}+9x.

$$-9x\left(8x-1\right)+8x-1$$

Factor out common term 8x-1 by using distributive property.

$$\left(8x-1\right)\left(-9x+1\right)$$

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.

$$-72x^{2}+17x-1=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-72\right)\left(-1\right)}}{2\left(-72\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-72\right)\left(-1\right)}}{2\left(-72\right)}$$

Multiply -4 times -72.

$$x=\frac{-17±\sqrt{289+288\left(-1\right)}}{2\left(-72\right)}$$

Multiply 288 times -1.

$$x=\frac{-17±\sqrt{289-288}}{2\left(-72\right)}$$

Add 289 to -288.

$$x=\frac{-17±\sqrt{1}}{2\left(-72\right)}$$

Take the square root of 1.

$$x=\frac{-17±1}{2\left(-72\right)}$$

Multiply 2 times -72.

$$x=\frac{-17±1}{-144}$$

Now solve the equation x=\frac{-17±1}{-144} when ± is plus. Add -17 to 1.

$$x=-\frac{16}{-144}$$

Reduce the fraction \frac{-16}{-144} to lowest terms by extracting and canceling out 16.

$$x=\frac{1}{9}$$

Now solve the equation x=\frac{-17±1}{-144} when ± is minus. Subtract 1 from -17.

$$x=-\frac{18}{-144}$$

Reduce the fraction \frac{-18}{-144} to lowest terms by extracting and canceling out 18.

$$x=\frac{1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{9} for x_{1} and \frac{1}{8} for x_{2}.

$$-72x^{2}+17x-1=-72\left(x-\frac{1}{9}\right)\left(x-\frac{1}{8}\right)$$

Subtract \frac{1}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}+17x-1=-72\times \frac{-9x+1}{-9}\left(x-\frac{1}{8}\right)$$

Subtract \frac{1}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}+17x-1=-72\times \frac{-9x+1}{-9}\times \frac{-8x+1}{-8}$$

Multiply \frac{-9x+1}{-9} times \frac{-8x+1}{-8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-72x^{2}+17x-1=-72\times \frac{\left(-9x+1\right)\left(-8x+1\right)}{-9\left(-8\right)}$$

Multiply -9 times -8.

$$-72x^{2}+17x-1=-72\times \frac{\left(-9x+1\right)\left(-8x+1\right)}{72}$$

Cancel out 72, the greatest common factor in -72 and 72.

$$-72x^{2}+17x-1=-\left(-9x+1\right)\left(-8x+1\right)$$

Evaluate

$$\left(1-9x\right)\left(8x-1\right)$$