Solve 5y^2+27y+10

$$5 y ^ { 2 } + 27 y + 10$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5y^{2}+ay+by+10. To find a and b, set up a system to be solved.

$$a+b=27 ab=5\times 10=50$$

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 50.

$$1,50 2,25 5,10$$

Calculate the sum for each pair.

$$1+50=51 2+25=27 5+10=15$$

The solution is the pair that gives sum 27.

$$a=2 b=25$$

Rewrite 5y^{2}+27y+10 as \left(5y^{2}+2y\right)+\left(25y+10\right).

$$\left(5y^{2}+2y\right)+\left(25y+10\right)$$

Factor out y in the first and 5 in the second group.

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

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

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

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.

$$5y^{2}+27y+10=0$$

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.

$$y=\frac{-27±\sqrt{27^{2}-4\times 5\times 10}}{2\times 5}$$

Square 27.

$$y=\frac{-27±\sqrt{729-4\times 5\times 10}}{2\times 5}$$

Multiply -4 times 5.

$$y=\frac{-27±\sqrt{729-20\times 10}}{2\times 5}$$

Multiply -20 times 10.

$$y=\frac{-27±\sqrt{729-200}}{2\times 5}$$

Add 729 to -200.

$$y=\frac{-27±\sqrt{529}}{2\times 5}$$

Take the square root of 529.

$$y=\frac{-27±23}{2\times 5}$$

Multiply 2 times 5.

$$y=\frac{-27±23}{10}$$

Now solve the equation y=\frac{-27±23}{10} when ± is plus. Add -27 to 23.

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

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

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

Now solve the equation y=\frac{-27±23}{10} when ± is minus. Subtract 23 from -27.

$$y=-\frac{50}{10}$$

Divide -50 by 10.

$$y=-5$$

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 -5 for x_{2}.

$$5y^{2}+27y+10=5\left(y-\left(-\frac{2}{5}\right)\right)\left(y-\left(-5\right)\right)$$

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

$$5y^{2}+27y+10=5\left(y+\frac{2}{5}\right)\left(y+5\right)$$

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

$$5y^{2}+27y+10=5\times \frac{5y+2}{5}\left(y+5\right)$$

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

$$5y^{2}+27y+10=\left(5y+2\right)\left(y+5\right)$$

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

$$x ^ 2 +\frac{27}{5}x +2 = 0$$

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

$$r + s = -\frac{27}{5} rs = 2$$

Two numbers r and s sum up to -\frac{27}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{27}{5} = -\frac{27}{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>

$$r = -\frac{27}{10} - u s = -\frac{27}{10} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(-\frac{27}{10} - u) (-\frac{27}{10} + u) = 2$$

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

$$\frac{729}{100} - u^2 = 2$$

Simplify the expression by subtracting \frac{729}{100} on both sides

$$-u^2 = 2-\frac{729}{100} = -\frac{529}{100}$$

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

$$u^2 = \frac{529}{100} u = \pm\sqrt{\frac{529}{100}} = \pm \frac{23}{10}$$

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

$$r =-\frac{27}{10} - \frac{23}{10} = -5 s = -\frac{27}{10} + \frac{23}{10} = -0.400$$

Evaluate

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