Solve -9z^2+71z+8

$$- 9 z ^ { 2 } + 71 z + 8$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -9z^{2}+az+bz+8. To find a and b, set up a system to be solved.

$$a+b=71 ab=-9\times 8=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 71.

$$a=72 b=-1$$

Rewrite -9z^{2}+71z+8 as \left(-9z^{2}+72z\right)+\left(-z+8\right).

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

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

$$9z\left(-z+8\right)-z+8$$

Factor out common term -z+8 by using distributive property.

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

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.

$$-9z^{2}+71z+8=0$$

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.

$$z=\frac{-71±\sqrt{71^{2}-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Square 71.

$$z=\frac{-71±\sqrt{5041-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$z=\frac{-71±\sqrt{5041+36\times 8}}{2\left(-9\right)}$$

Multiply 36 times 8.

$$z=\frac{-71±\sqrt{5041+288}}{2\left(-9\right)}$$

Add 5041 to 288.

$$z=\frac{-71±\sqrt{5329}}{2\left(-9\right)}$$

Take the square root of 5329.

$$z=\frac{-71±73}{2\left(-9\right)}$$

Multiply 2 times -9.

$$z=\frac{-71±73}{-18}$$

Now solve the equation z=\frac{-71±73}{-18} when ± is plus. Add -71 to 73.

$$z=\frac{2}{-18}$$

Reduce the fraction \frac{2}{-18} to lowest terms by extracting and canceling out 2.

$$z=-\frac{1}{9}$$

Now solve the equation z=\frac{-71±73}{-18} when ± is minus. Subtract 73 from -71.

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

Divide -144 by -18.

$$z=8$$

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 8 for x_{2}.

$$-9z^{2}+71z+8=-9\left(z-\left(-\frac{1}{9}\right)\right)\left(z-8\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-9z^{2}+71z+8=-9\left(z+\frac{1}{9}\right)\left(z-8\right)$$

Add \frac{1}{9} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-9z^{2}+71z+8=-9\times \frac{-9z-1}{-9}\left(z-8\right)$$

Cancel out 9, the greatest common factor in -9 and 9.

$$-9z^{2}+71z+8=\left(-9z-1\right)\left(z-8\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -\frac{71}{9}x -\frac{8}{9} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{71}{9} rs = -\frac{8}{9}$$

Two numbers r and s sum up to \frac{71}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{71}{9} = \frac{71}{18}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{71}{18} - u s = \frac{71}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{9}

$$(\frac{71}{18} - u) (\frac{71}{18} + u) = -\frac{8}{9}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{5041}{324} - u^2 = -\frac{8}{9}$$

Simplify the expression by subtracting \frac{5041}{324} on both sides

$$-u^2 = -\frac{8}{9}-\frac{5041}{324} = -\frac{5329}{324}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{5329}{324} u = \pm\sqrt{\frac{5329}{324}} = \pm \frac{73}{18}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{71}{18} - \frac{73}{18} = -0.111 s = \frac{71}{18} + \frac{73}{18} = 8$$

Evaluate

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