Solve 2z^2+15z-8

$$2 z ^ { 2 } + 15 z - 8$$

Factor

$$\left(2z-1\right)\left(z+8\right)$$

Solution Steps

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

$$a+b=15 ab=2\left(-8\right)=-16$$

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 -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 15.

$$a=-1 b=16$$

Rewrite 2z^{2}+15z-8 as \left(2z^{2}-z\right)+\left(16z-8\right).

$$\left(2z^{2}-z\right)+\left(16z-8\right)$$

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

$$z\left(2z-1\right)+8\left(2z-1\right)$$

Factor out common term 2z-1 by using distributive property.

$$\left(2z-1\right)\left(z+8\right)$$

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.

$$2z^{2}+15z-8=0$$

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.

$$z=\frac{-15±\sqrt{15^{2}-4\times 2\left(-8\right)}}{2\times 2}$$

Square 15.

$$z=\frac{-15±\sqrt{225-4\times 2\left(-8\right)}}{2\times 2}$$

Multiply -4 times 2.

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

Multiply -8 times -8.

$$z=\frac{-15±\sqrt{225+64}}{2\times 2}$$

Add 225 to 64.

$$z=\frac{-15±\sqrt{289}}{2\times 2}$$

Take the square root of 289.

$$z=\frac{-15±17}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{-15±17}{4}$$

Now solve the equation z=\frac{-15±17}{4} when ± is plus. Add -15 to 17.

$$z=\frac{2}{4}$$

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

$$z=\frac{1}{2}$$

Now solve the equation z=\frac{-15±17}{4} when ± is minus. Subtract 17 from -15.

$$z=-\frac{32}{4}$$

Divide -32 by 4.

$$z=-8$$

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

$$2z^{2}+15z-8=2\left(z-\frac{1}{2}\right)\left(z-\left(-8\right)\right)$$

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

$$2z^{2}+15z-8=2\left(z-\frac{1}{2}\right)\left(z+8\right)$$

Subtract \frac{1}{2} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2z^{2}+15z-8=2\times \frac{2z-1}{2}\left(z+8\right)$$

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

$$2z^{2}+15z-8=\left(2z-1\right)\left(z+8\right)$$

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.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 +\frac{15}{2}x -4 = 0$$

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

$$r + s = -\frac{15}{2} rs = -4$$

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 = -4

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

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

$$\frac{225}{16} - u^2 = -4$$

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

$$-u^2 = -4-\frac{225}{16} = -\frac{289}{16}$$

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

$$u^2 = \frac{289}{16} u = \pm\sqrt{\frac{289}{16}} = \pm \frac{17}{4}$$

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

$$r =-\frac{15}{4} - \frac{17}{4} = -8 s = -\frac{15}{4} + \frac{17}{4} = 0.500$$

Evaluate

$$\left(2z-1\right)\left(z+8\right)$$