Solve 5x^2+29x+20

$$5 x ^ { 2 } + 29 x + 20$$

Factor

$$\left(x+5\right)\left(5x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=29 ab=5\times 20=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 29.

$$a=4 b=25$$

Rewrite 5x^{2}+29x+20 as \left(5x^{2}+4x\right)+\left(25x+20\right).

$$\left(5x^{2}+4x\right)+\left(25x+20\right)$$

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

$$x\left(5x+4\right)+5\left(5x+4\right)$$

Factor out common term 5x+4 by using distributive property.

$$\left(5x+4\right)\left(x+5\right)$$

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.

$$5x^{2}+29x+20=0$$

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.

$$x=\frac{-29±\sqrt{29^{2}-4\times 5\times 20}}{2\times 5}$$

Square 29.

$$x=\frac{-29±\sqrt{841-4\times 5\times 20}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-29±\sqrt{841-20\times 20}}{2\times 5}$$

Multiply -20 times 20.

$$x=\frac{-29±\sqrt{841-400}}{2\times 5}$$

Add 841 to -400.

$$x=\frac{-29±\sqrt{441}}{2\times 5}$$

Take the square root of 441.

$$x=\frac{-29±21}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-29±21}{10}$$

Now solve the equation x=\frac{-29±21}{10} when ± is plus. Add -29 to 21.

$$x=-\frac{8}{10}$$

Reduce the fraction \frac{-8}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{4}{5}$$

Now solve the equation x=\frac{-29±21}{10} when ± is minus. Subtract 21 from -29.

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

Divide -50 by 10.

$$x=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{5} for x_{1} and -5 for x_{2}.

$$5x^{2}+29x+20=5\left(x-\left(-\frac{4}{5}\right)\right)\left(x-\left(-5\right)\right)$$

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

$$5x^{2}+29x+20=5\left(x+\frac{4}{5}\right)\left(x+5\right)$$

Add \frac{4}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}+29x+20=5\times \frac{5x+4}{5}\left(x+5\right)$$

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

$$5x^{2}+29x+20=\left(5x+4\right)\left(x+5\right)$$

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

$$x ^ 2 +\frac{29}{5}x +4 = 0$$

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

$$r + s = -\frac{29}{5} rs = 4$$

Two numbers r and s sum up to -\frac{29}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{29}{5} = -\frac{29}{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>

$$r = -\frac{29}{10} - u s = -\frac{29}{10} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4

$$(-\frac{29}{10} - u) (-\frac{29}{10} + u) = 4$$

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

$$\frac{841}{100} - u^2 = 4$$

Simplify the expression by subtracting \frac{841}{100} on both sides

$$-u^2 = 4-\frac{841}{100} = -\frac{441}{100}$$

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

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

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

$$r =-\frac{29}{10} - \frac{21}{10} = -5 s = -\frac{29}{10} + \frac{21}{10} = -0.800$$

Evaluate

$$\left(x+5\right)\left(5x+4\right)$$