Solve 2z^2-15z+18

$$2 z ^ { 2 } - 15 z + 18$$

Factor

$$\left(z-6\right)\left(2z-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2z^{2}+az+bz+18. To find a and b, set up a system to be solved.

$$a+b=-15 ab=2\times 18=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -15.

$$a=-12 b=-3$$

Rewrite 2z^{2}-15z+18 as \left(2z^{2}-12z\right)+\left(-3z+18\right).

$$\left(2z^{2}-12z\right)+\left(-3z+18\right)$$

Factor out 2z in the first and -3 in the second group.

$$2z\left(z-6\right)-3\left(z-6\right)$$

Factor out common term z-6 by using distributive property.

$$\left(z-6\right)\left(2z-3\right)$$

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.

$$2z^{2}-15z+18=0$$

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.

$$z=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 2\times 18}}{2\times 2}$$

Square -15.

$$z=\frac{-\left(-15\right)±\sqrt{225-4\times 2\times 18}}{2\times 2}$$

Multiply -4 times 2.

$$z=\frac{-\left(-15\right)±\sqrt{225-8\times 18}}{2\times 2}$$

Multiply -8 times 18.

$$z=\frac{-\left(-15\right)±\sqrt{225-144}}{2\times 2}$$

Add 225 to -144.

$$z=\frac{-\left(-15\right)±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$z=\frac{-\left(-15\right)±9}{2\times 2}$$

The opposite of -15 is 15.

$$z=\frac{15±9}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{15±9}{4}$$

Now solve the equation z=\frac{15±9}{4} when ± is plus. Add 15 to 9.

$$z=\frac{24}{4}$$

Divide 24 by 4.

$$z=6$$

Now solve the equation z=\frac{15±9}{4} when ± is minus. Subtract 9 from 15.

$$z=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$z=\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and \frac{3}{2} for x_{2}.

$$2z^{2}-15z+18=2\left(z-6\right)\left(z-\frac{3}{2}\right)$$

Subtract \frac{3}{2} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2z^{2}-15z+18=2\left(z-6\right)\times \frac{2z-3}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2z^{2}-15z+18=\left(z-6\right)\left(2z-3\right)$$

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 2

$$x ^ 2 -\frac{15}{2}x +9 = 0$$

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

$$r + s = \frac{15}{2} rs = 9$$

Two numbers r and s sum up to \frac{15}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{15}{2} = \frac{15}{4}. 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>

$$r = \frac{15}{4} - u s = \frac{15}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 9

$$(\frac{15}{4} - u) (\frac{15}{4} + u) = 9$$

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

$$\frac{225}{16} - u^2 = 9$$

Simplify the expression by subtracting \frac{225}{16} on both sides

$$-u^2 = 9-\frac{225}{16} = -\frac{81}{16}$$

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

$$u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}$$

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

$$r =\frac{15}{4} - \frac{9}{4} = 1.500 s = \frac{15}{4} + \frac{9}{4} = 6$$

Evaluate

$$\left(z-6\right)\left(2z-3\right)$$