Solve 5z^2+7z+2

$$5 z ^ { 2 } + 7 z + 2$$

Factor

$$\left(z+1\right)\left(5z+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5z^{2}+az+bz+2. To find a and b, set up a system to be solved.

$$a+b=7 ab=5\times 2=10$$

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 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 7.

$$a=2 b=5$$

Rewrite 5z^{2}+7z+2 as \left(5z^{2}+2z\right)+\left(5z+2\right).

$$\left(5z^{2}+2z\right)+\left(5z+2\right)$$

Factor out z in 5z^{2}+2z.

$$z\left(5z+2\right)+5z+2$$

Factor out common term 5z+2 by using distributive property.

$$\left(5z+2\right)\left(z+1\right)$$

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.

$$5z^{2}+7z+2=0$$

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.

$$z=\frac{-7±\sqrt{7^{2}-4\times 5\times 2}}{2\times 5}$$

Square 7.

$$z=\frac{-7±\sqrt{49-4\times 5\times 2}}{2\times 5}$$

Multiply -4 times 5.

$$z=\frac{-7±\sqrt{49-20\times 2}}{2\times 5}$$

Multiply -20 times 2.

$$z=\frac{-7±\sqrt{49-40}}{2\times 5}$$

Add 49 to -40.

$$z=\frac{-7±\sqrt{9}}{2\times 5}$$

Take the square root of 9.

$$z=\frac{-7±3}{2\times 5}$$

Multiply 2 times 5.

$$z=\frac{-7±3}{10}$$

Now solve the equation z=\frac{-7±3}{10} when ± is plus. Add -7 to 3.

$$z=-\frac{4}{10}$$

Reduce the fraction \frac{-4}{10} to lowest terms by extracting and canceling out 2.

$$z=-\frac{2}{5}$$

Now solve the equation z=\frac{-7±3}{10} when ± is minus. Subtract 3 from -7.

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

Divide -10 by 10.

$$z=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{5} for x_{1} and -1 for x_{2}.

$$5z^{2}+7z+2=5\left(z-\left(-\frac{2}{5}\right)\right)\left(z-\left(-1\right)\right)$$

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

$$5z^{2}+7z+2=5\left(z+\frac{2}{5}\right)\left(z+1\right)$$

Add \frac{2}{5} to z by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5z^{2}+7z+2=5\times \frac{5z+2}{5}\left(z+1\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5z^{2}+7z+2=\left(5z+2\right)\left(z+1\right)$$

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 5

$$x ^ 2 +\frac{7}{5}x +\frac{2}{5} = 0$$

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

$$r + s = -\frac{7}{5} rs = \frac{2}{5}$$

Two numbers r and s sum up to -\frac{7}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{5} = -\frac{7}{10}. 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>

$$r = -\frac{7}{10} - u s = -\frac{7}{10} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2}{5}

$$(-\frac{7}{10} - u) (-\frac{7}{10} + u) = \frac{2}{5}$$

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

$$\frac{49}{100} - u^2 = \frac{2}{5}$$

Simplify the expression by subtracting \frac{49}{100} on both sides

$$-u^2 = \frac{2}{5}-\frac{49}{100} = -\frac{9}{100}$$

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

$$u^2 = \frac{9}{100} u = \pm\sqrt{\frac{9}{100}} = \pm \frac{3}{10}$$

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

$$r =-\frac{7}{10} - \frac{3}{10} = -1.000 s = -\frac{7}{10} + \frac{3}{10} = -0.400$$

Evaluate

$$\left(z+1\right)\left(5z+2\right)$$