Solve 8z^2-11z+3

$$8 z ^ { 2 } - 11 z + 3$$

Factor

$$\left(z-1\right)\left(8z-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8z^{2}+az+bz+3. To find a and b, set up a system to be solved.

$$a+b=-11 ab=8\times 3=24$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 8z^{2}-11z+3 as \left(8z^{2}-8z\right)+\left(-3z+3\right).

$$\left(8z^{2}-8z\right)+\left(-3z+3\right)$$

Factor out 8z in the first and -3 in the second group.

$$8z\left(z-1\right)-3\left(z-1\right)$$

Factor out common term z-1 by using distributive property.

$$\left(z-1\right)\left(8z-3\right)$$

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.

$$8z^{2}-11z+3=0$$

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.

$$z=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 8\times 3}}{2\times 8}$$

Square -11.

$$z=\frac{-\left(-11\right)±\sqrt{121-4\times 8\times 3}}{2\times 8}$$

Multiply -4 times 8.

$$z=\frac{-\left(-11\right)±\sqrt{121-32\times 3}}{2\times 8}$$

Multiply -32 times 3.

$$z=\frac{-\left(-11\right)±\sqrt{121-96}}{2\times 8}$$

Add 121 to -96.

$$z=\frac{-\left(-11\right)±\sqrt{25}}{2\times 8}$$

Take the square root of 25.

$$z=\frac{-\left(-11\right)±5}{2\times 8}$$

The opposite of -11 is 11.

$$z=\frac{11±5}{2\times 8}$$

Multiply 2 times 8.

$$z=\frac{11±5}{16}$$

Now solve the equation z=\frac{11±5}{16} when ± is plus. Add 11 to 5.

$$z=\frac{16}{16}$$

Divide 16 by 16.

$$z=1$$

Now solve the equation z=\frac{11±5}{16} when ± is minus. Subtract 5 from 11.

$$z=\frac{6}{16}$$

Reduce the fraction \frac{6}{16} to lowest terms by extracting and canceling out 2.

$$z=\frac{3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and \frac{3}{8} for x_{2}.

$$8z^{2}-11z+3=8\left(z-1\right)\left(z-\frac{3}{8}\right)$$

Subtract \frac{3}{8} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8z^{2}-11z+3=8\left(z-1\right)\times \frac{8z-3}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8z^{2}-11z+3=\left(z-1\right)\left(8z-3\right)$$

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.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -\frac{11}{8}x +\frac{3}{8} = 0$$

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

$$r + s = \frac{11}{8} rs = \frac{3}{8}$$

Two numbers r and s sum up to \frac{11}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{8} = \frac{11}{16}. 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>

$$r = \frac{11}{16} - u s = \frac{11}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{8}

$$(\frac{11}{16} - u) (\frac{11}{16} + u) = \frac{3}{8}$$

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

$$\frac{121}{256} - u^2 = \frac{3}{8}$$

Simplify the expression by subtracting \frac{121}{256} on both sides

$$-u^2 = \frac{3}{8}-\frac{121}{256} = -\frac{25}{256}$$

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

$$u^2 = \frac{25}{256} u = \pm\sqrt{\frac{25}{256}} = \pm \frac{5}{16}$$

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

$$r =\frac{11}{16} - \frac{5}{16} = 0.375 s = \frac{11}{16} + \frac{5}{16} = 1$$

Evaluate

$$\left(z-1\right)\left(8z-3\right)$$