Solve 660x^2+524x+85

$$660 x ^ { 2 } + 524 x + 85$$

Factor

$$\left(22x+5\right)\left(30x+17\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 660x^{2}+ax+bx+85. To find a and b, set up a system to be solved.

$$a+b=524 ab=660\times 85=56100$$

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 56100.

$$1,56100 2,28050 3,18700 4,14025 5,11220 6,9350 10,5610 11,5100 12,4675 15,3740 17,3300 20,2805 22,2550 25,2244 30,1870 33,1700 34,1650 44,1275 50,1122 51,1100 55,1020 60,935 66,850 68,825 75,748 85,660 100,561 102,550 110,510 132,425 150,374 165,340 170,330 187,300 204,275 220,255$$

Calculate the sum for each pair.

$$1+56100=56101 2+28050=28052 3+18700=18703 4+14025=14029 5+11220=11225 6+9350=9356 10+5610=5620 11+5100=5111 12+4675=4687 15+3740=3755 17+3300=3317 20+2805=2825 22+2550=2572 25+2244=2269 30+1870=1900 33+1700=1733 34+1650=1684 44+1275=1319 50+1122=1172 51+1100=1151 55+1020=1075 60+935=995 66+850=916 68+825=893 75+748=823 85+660=745 100+561=661 102+550=652 110+510=620 132+425=557 150+374=524 165+340=505 170+330=500 187+300=487 204+275=479 220+255=475$$

The solution is the pair that gives sum 524.

$$a=150 b=374$$

Rewrite 660x^{2}+524x+85 as \left(660x^{2}+150x\right)+\left(374x+85\right).

$$\left(660x^{2}+150x\right)+\left(374x+85\right)$$

Factor out 30x in the first and 17 in the second group.

$$30x\left(22x+5\right)+17\left(22x+5\right)$$

Factor out common term 22x+5 by using distributive property.

$$\left(22x+5\right)\left(30x+17\right)$$

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.

$$660x^{2}+524x+85=0$$

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.

$$x=\frac{-524±\sqrt{524^{2}-4\times 660\times 85}}{2\times 660}$$

Square 524.

$$x=\frac{-524±\sqrt{274576-4\times 660\times 85}}{2\times 660}$$

Multiply -4 times 660.

$$x=\frac{-524±\sqrt{274576-2640\times 85}}{2\times 660}$$

Multiply -2640 times 85.

$$x=\frac{-524±\sqrt{274576-224400}}{2\times 660}$$

Add 274576 to -224400.

$$x=\frac{-524±\sqrt{50176}}{2\times 660}$$

Take the square root of 50176.

$$x=\frac{-524±224}{2\times 660}$$

Multiply 2 times 660.

$$x=\frac{-524±224}{1320}$$

Now solve the equation x=\frac{-524±224}{1320} when ± is plus. Add -524 to 224.

$$x=-\frac{300}{1320}$$

Reduce the fraction \frac{-300}{1320} to lowest terms by extracting and canceling out 60.

$$x=-\frac{5}{22}$$

Now solve the equation x=\frac{-524±224}{1320} when ± is minus. Subtract 224 from -524.

$$x=-\frac{748}{1320}$$

Reduce the fraction \frac{-748}{1320} to lowest terms by extracting and canceling out 44.

$$x=-\frac{17}{30}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{22} for x_{1} and -\frac{17}{30} for x_{2}.

$$660x^{2}+524x+85=660\left(x-\left(-\frac{5}{22}\right)\right)\left(x-\left(-\frac{17}{30}\right)\right)$$

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

$$660x^{2}+524x+85=660\left(x+\frac{5}{22}\right)\left(x+\frac{17}{30}\right)$$

Add \frac{5}{22} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$660x^{2}+524x+85=660\times \frac{22x+5}{22}\left(x+\frac{17}{30}\right)$$

Add \frac{17}{30} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$660x^{2}+524x+85=660\times \frac{22x+5}{22}\times \frac{30x+17}{30}$$

Multiply \frac{22x+5}{22} times \frac{30x+17}{30} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$660x^{2}+524x+85=660\times \frac{\left(22x+5\right)\left(30x+17\right)}{22\times 30}$$

Multiply 22 times 30.

$$660x^{2}+524x+85=660\times \frac{\left(22x+5\right)\left(30x+17\right)}{660}$$

Cancel out 660, the greatest common factor in 660 and 660.

$$660x^{2}+524x+85=\left(22x+5\right)\left(30x+17\right)$$

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 660

$$x ^ 2 +\frac{131}{165}x +\frac{17}{132} = 0$$

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

$$r + s = -\frac{131}{165} rs = \frac{17}{132}$$

Two numbers r and s sum up to -\frac{131}{165} exactly when the average of the two numbers is \frac{1}{2}*-\frac{131}{165} = -\frac{131}{330}. 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>

$$r = -\frac{131}{330} - u s = -\frac{131}{330} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{17}{132}

$$(-\frac{131}{330} - u) (-\frac{131}{330} + u) = \frac{17}{132}$$

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

$$\frac{17161}{108900} - u^2 = \frac{17}{132}$$

Simplify the expression by subtracting \frac{17161}{108900} on both sides

$$-u^2 = \frac{17}{132}-\frac{17161}{108900} = -\frac{784}{27225}$$

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

$$u^2 = \frac{784}{27225} u = \pm\sqrt{\frac{784}{27225}} = \pm \frac{28}{165}$$

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

$$r =-\frac{131}{330} - \frac{28}{165} = -0.567 s = -\frac{131}{330} + \frac{28}{165} = -0.227$$

Evaluate

$$\left(22x+5\right)\left(30x+17\right)$$