Solve 21z^2+8z-4

$$21 z ^ { 2 } + 8 z - 4$$

Factor

$$\left(7z-2\right)\left(3z+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 21z^{2}+az+bz-4. To find a and b, set up a system to be solved.

$$a+b=8 ab=21\left(-4\right)=-84$$

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 -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 8.

$$a=-6 b=14$$

Rewrite 21z^{2}+8z-4 as \left(21z^{2}-6z\right)+\left(14z-4\right).

$$\left(21z^{2}-6z\right)+\left(14z-4\right)$$

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

$$3z\left(7z-2\right)+2\left(7z-2\right)$$

Factor out common term 7z-2 by using distributive property.

$$\left(7z-2\right)\left(3z+2\right)$$

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.

$$21z^{2}+8z-4=0$$

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.

$$z=\frac{-8±\sqrt{8^{2}-4\times 21\left(-4\right)}}{2\times 21}$$

Square 8.

$$z=\frac{-8±\sqrt{64-4\times 21\left(-4\right)}}{2\times 21}$$

Multiply -4 times 21.

$$z=\frac{-8±\sqrt{64-84\left(-4\right)}}{2\times 21}$$

Multiply -84 times -4.

$$z=\frac{-8±\sqrt{64+336}}{2\times 21}$$

Add 64 to 336.

$$z=\frac{-8±\sqrt{400}}{2\times 21}$$

Take the square root of 400.

$$z=\frac{-8±20}{2\times 21}$$

Multiply 2 times 21.

$$z=\frac{-8±20}{42}$$

Now solve the equation z=\frac{-8±20}{42} when ± is plus. Add -8 to 20.

$$z=\frac{12}{42}$$

Reduce the fraction \frac{12}{42} to lowest terms by extracting and canceling out 6.

$$z=\frac{2}{7}$$

Now solve the equation z=\frac{-8±20}{42} when ± is minus. Subtract 20 from -8.

$$z=-\frac{28}{42}$$

Reduce the fraction \frac{-28}{42} to lowest terms by extracting and canceling out 14.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{7} for x_{1} and -\frac{2}{3} for x_{2}.

$$21z^{2}+8z-4=21\left(z-\frac{2}{7}\right)\left(z-\left(-\frac{2}{3}\right)\right)$$

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

$$21z^{2}+8z-4=21\left(z-\frac{2}{7}\right)\left(z+\frac{2}{3}\right)$$

Subtract \frac{2}{7} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$21z^{2}+8z-4=21\times \frac{7z-2}{7}\left(z+\frac{2}{3}\right)$$

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

$$21z^{2}+8z-4=21\times \frac{7z-2}{7}\times \frac{3z+2}{3}$$

Multiply \frac{7z-2}{7} times \frac{3z+2}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$21z^{2}+8z-4=21\times \frac{\left(7z-2\right)\left(3z+2\right)}{7\times 3}$$

Multiply 7 times 3.

$$21z^{2}+8z-4=21\times \frac{\left(7z-2\right)\left(3z+2\right)}{21}$$

Cancel out 21, the greatest common factor in 21 and 21.

$$21z^{2}+8z-4=\left(7z-2\right)\left(3z+2\right)$$

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 21

$$x ^ 2 +\frac{8}{21}x -\frac{4}{21} = 0$$

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

$$r + s = -\frac{8}{21} rs = -\frac{4}{21}$$

Two numbers r and s sum up to -\frac{8}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{21} = -\frac{4}{21}. 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>

$$r = -\frac{4}{21} - u s = -\frac{4}{21} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{21}

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

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

$$\frac{16}{441} - u^2 = -\frac{4}{21}$$

Simplify the expression by subtracting \frac{16}{441} on both sides

$$-u^2 = -\frac{4}{21}-\frac{16}{441} = -\frac{100}{441}$$

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

$$u^2 = \frac{100}{441} u = \pm\sqrt{\frac{100}{441}} = \pm \frac{10}{21}$$

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

$$r =-\frac{4}{21} - \frac{10}{21} = -0.667 s = -\frac{4}{21} + \frac{10}{21} = 0.286$$

Evaluate

$$\left(7z-2\right)\left(3z+2\right)$$