Solve 10z^2+21z+2

$$10 z ^ { 2 } + 21 z + 2$$

Factor

$$\left(z+2\right)\left(10z+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 10z^{2}+az+bz+2. To find a and b, set up a system to be solved.

$$a+b=21 ab=10\times 2=20$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 21.

$$a=1 b=20$$

Rewrite 10z^{2}+21z+2 as \left(10z^{2}+z\right)+\left(20z+2\right).

$$\left(10z^{2}+z\right)+\left(20z+2\right)$$

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

$$z\left(10z+1\right)+2\left(10z+1\right)$$

Factor out common term 10z+1 by using distributive property.

$$\left(10z+1\right)\left(z+2\right)$$

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.

$$10z^{2}+21z+2=0$$

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.

$$z=\frac{-21±\sqrt{21^{2}-4\times 10\times 2}}{2\times 10}$$

Square 21.

$$z=\frac{-21±\sqrt{441-4\times 10\times 2}}{2\times 10}$$

Multiply -4 times 10.

$$z=\frac{-21±\sqrt{441-40\times 2}}{2\times 10}$$

Multiply -40 times 2.

$$z=\frac{-21±\sqrt{441-80}}{2\times 10}$$

Add 441 to -80.

$$z=\frac{-21±\sqrt{361}}{2\times 10}$$

Take the square root of 361.

$$z=\frac{-21±19}{2\times 10}$$

Multiply 2 times 10.

$$z=\frac{-21±19}{20}$$

Now solve the equation z=\frac{-21±19}{20} when ± is plus. Add -21 to 19.

$$z=-\frac{2}{20}$$

Reduce the fraction \frac{-2}{20} to lowest terms by extracting and canceling out 2.

$$z=-\frac{1}{10}$$

Now solve the equation z=\frac{-21±19}{20} when ± is minus. Subtract 19 from -21.

$$z=-\frac{40}{20}$$

Divide -40 by 20.

$$z=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{10} for x_{1} and -2 for x_{2}.

$$10z^{2}+21z+2=10\left(z-\left(-\frac{1}{10}\right)\right)\left(z-\left(-2\right)\right)$$

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

$$10z^{2}+21z+2=10\left(z+\frac{1}{10}\right)\left(z+2\right)$$

Add \frac{1}{10} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$10z^{2}+21z+2=10\times \frac{10z+1}{10}\left(z+2\right)$$

Cancel out 10, the greatest common factor in 10 and 10.

$$10z^{2}+21z+2=\left(10z+1\right)\left(z+2\right)$$

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 10

$$x ^ 2 +\frac{21}{10}x +\frac{1}{5} = 0$$

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

$$r + s = -\frac{21}{10} rs = \frac{1}{5}$$

Two numbers r and s sum up to -\frac{21}{10} exactly when the average of the two numbers is \frac{1}{2}*-\frac{21}{10} = -\frac{21}{20}. 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>

$$r = -\frac{21}{20} - u s = -\frac{21}{20} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{5}

$$(-\frac{21}{20} - u) (-\frac{21}{20} + u) = \frac{1}{5}$$

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

$$\frac{441}{400} - u^2 = \frac{1}{5}$$

Simplify the expression by subtracting \frac{441}{400} on both sides

$$-u^2 = \frac{1}{5}-\frac{441}{400} = -\frac{361}{400}$$

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

$$u^2 = \frac{361}{400} u = \pm\sqrt{\frac{361}{400}} = \pm \frac{19}{20}$$

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

$$r =-\frac{21}{20} - \frac{19}{20} = -2 s = -\frac{21}{20} + \frac{19}{20} = -0.100$$

Evaluate

$$\left(z+2\right)\left(10z+1\right)$$