Solve 7z^2+22z+3

$$7 z ^ { 2 } + 22 z + 3$$

Factor

$$\left(z+3\right)\left(7z+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7z^{2}+az+bz+3. To find a and b, set up a system to be solved.

$$a+b=22 ab=7\times 3=21$$

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 21.

$$1,21 3,7$$

Calculate the sum for each pair.

$$1+21=22 3+7=10$$

The solution is the pair that gives sum 22.

$$a=1 b=21$$

Rewrite 7z^{2}+22z+3 as \left(7z^{2}+z\right)+\left(21z+3\right).

$$\left(7z^{2}+z\right)+\left(21z+3\right)$$

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

$$z\left(7z+1\right)+3\left(7z+1\right)$$

Factor out common term 7z+1 by using distributive property.

$$\left(7z+1\right)\left(z+3\right)$$

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.

$$7z^{2}+22z+3=0$$

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.

$$z=\frac{-22±\sqrt{22^{2}-4\times 7\times 3}}{2\times 7}$$

Square 22.

$$z=\frac{-22±\sqrt{484-4\times 7\times 3}}{2\times 7}$$

Multiply -4 times 7.

$$z=\frac{-22±\sqrt{484-28\times 3}}{2\times 7}$$

Multiply -28 times 3.

$$z=\frac{-22±\sqrt{484-84}}{2\times 7}$$

Add 484 to -84.

$$z=\frac{-22±\sqrt{400}}{2\times 7}$$

Take the square root of 400.

$$z=\frac{-22±20}{2\times 7}$$

Multiply 2 times 7.

$$z=\frac{-22±20}{14}$$

Now solve the equation z=\frac{-22±20}{14} when ± is plus. Add -22 to 20.

$$z=-\frac{2}{14}$$

Reduce the fraction \frac{-2}{14} to lowest terms by extracting and canceling out 2.

$$z=-\frac{1}{7}$$

Now solve the equation z=\frac{-22±20}{14} when ± is minus. Subtract 20 from -22.

$$z=-\frac{42}{14}$$

Divide -42 by 14.

$$z=-3$$

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

$$7z^{2}+22z+3=7\left(z-\left(-\frac{1}{7}\right)\right)\left(z-\left(-3\right)\right)$$

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

$$7z^{2}+22z+3=7\left(z+\frac{1}{7}\right)\left(z+3\right)$$

Add \frac{1}{7} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7z^{2}+22z+3=7\times \frac{7z+1}{7}\left(z+3\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7z^{2}+22z+3=\left(7z+1\right)\left(z+3\right)$$

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 7

$$x ^ 2 +\frac{22}{7}x +\frac{3}{7} = 0$$

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

$$r + s = -\frac{22}{7} rs = \frac{3}{7}$$

Two numbers r and s sum up to -\frac{22}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{22}{7} = -\frac{11}{7}. 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>

$$r = -\frac{11}{7} - u s = -\frac{11}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{7}

$$(-\frac{11}{7} - u) (-\frac{11}{7} + u) = \frac{3}{7}$$

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

$$\frac{121}{49} - u^2 = \frac{3}{7}$$

Simplify the expression by subtracting \frac{121}{49} on both sides

$$-u^2 = \frac{3}{7}-\frac{121}{49} = -\frac{100}{49}$$

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

$$u^2 = \frac{100}{49} u = \pm\sqrt{\frac{100}{49}} = \pm \frac{10}{7}$$

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

$$r =-\frac{11}{7} - \frac{10}{7} = -3 s = -\frac{11}{7} + \frac{10}{7} = -0.143$$

Evaluate

$$\left(z+3\right)\left(7z+1\right)$$