Solve 2z^2+19z-21

$$2 z ^ { 2 } + 19 z - 21$$

Factor

$$\left(z-1\right)\left(2z+21\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2z^{2}+az+bz-21. To find a and b, set up a system to be solved.

$$a+b=19 ab=2\left(-21\right)=-42$$

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 -42.

$$-1,42 -2,21 -3,14 -6,7$$

Calculate the sum for each pair.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

The solution is the pair that gives sum 19.

$$a=-2 b=21$$

Rewrite 2z^{2}+19z-21 as \left(2z^{2}-2z\right)+\left(21z-21\right).

$$\left(2z^{2}-2z\right)+\left(21z-21\right)$$

Factor out 2z in the first and 21 in the second group.

$$2z\left(z-1\right)+21\left(z-1\right)$$

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

$$\left(z-1\right)\left(2z+21\right)$$

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.

$$2z^{2}+19z-21=0$$

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.

$$z=\frac{-19±\sqrt{19^{2}-4\times 2\left(-21\right)}}{2\times 2}$$

Square 19.

$$z=\frac{-19±\sqrt{361-4\times 2\left(-21\right)}}{2\times 2}$$

Multiply -4 times 2.

$$z=\frac{-19±\sqrt{361-8\left(-21\right)}}{2\times 2}$$

Multiply -8 times -21.

$$z=\frac{-19±\sqrt{361+168}}{2\times 2}$$

Add 361 to 168.

$$z=\frac{-19±\sqrt{529}}{2\times 2}$$

Take the square root of 529.

$$z=\frac{-19±23}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{-19±23}{4}$$

Now solve the equation z=\frac{-19±23}{4} when ± is plus. Add -19 to 23.

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

Divide 4 by 4.

$$z=1$$

Now solve the equation z=\frac{-19±23}{4} when ± is minus. Subtract 23 from -19.

$$z=-\frac{42}{4}$$

Reduce the fraction \frac{-42}{4} to lowest terms by extracting and canceling out 2.

$$z=-\frac{21}{2}$$

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{21}{2} for x_{2}.

$$2z^{2}+19z-21=2\left(z-1\right)\left(z-\left(-\frac{21}{2}\right)\right)$$

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

$$2z^{2}+19z-21=2\left(z-1\right)\left(z+\frac{21}{2}\right)$$

Add \frac{21}{2} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2z^{2}+19z-21=2\left(z-1\right)\times \frac{2z+21}{2}$$

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

$$2z^{2}+19z-21=\left(z-1\right)\left(2z+21\right)$$

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

$$x ^ 2 +\frac{19}{2}x -\frac{21}{2} = 0$$

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

$$r + s = -\frac{19}{2} rs = -\frac{21}{2}$$

Two numbers r and s sum up to -\frac{19}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{19}{2} = -\frac{19}{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>

$$r = -\frac{19}{4} - u s = -\frac{19}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{2}

$$(-\frac{19}{4} - u) (-\frac{19}{4} + u) = -\frac{21}{2}$$

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

$$\frac{361}{16} - u^2 = -\frac{21}{2}$$

Simplify the expression by subtracting \frac{361}{16} on both sides

$$-u^2 = -\frac{21}{2}-\frac{361}{16} = -\frac{529}{16}$$

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

$$u^2 = \frac{529}{16} u = \pm\sqrt{\frac{529}{16}} = \pm \frac{23}{4}$$

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

$$r =-\frac{19}{4} - \frac{23}{4} = -10.500 s = -\frac{19}{4} + \frac{23}{4} = 1$$

Evaluate

$$\left(z-1\right)\left(2z+21\right)$$