Solve 624x^2+1890x+1176

$$624 { x }^{ 2 } +1890x+1176$$

Factor

$$6\left(8x+7\right)\left(13x+28\right)$$

Solution Steps

Factor out 6.

$$6\left(104x^{2}+315x+196\right)$$

Consider 104x^{2}+315x+196. Factor the expression by grouping. First, the expression needs to be rewritten as 104x^{2}+ax+bx+196. To find a and b, set up a system to be solved.

$$a+b=315 ab=104\times 196=20384$$

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 20384.

$$1,20384 2,10192 4,5096 7,2912 8,2548 13,1568 14,1456 16,1274 26,784 28,728 32,637 49,416 52,392 56,364 91,224 98,208 104,196 112,182$$

Calculate the sum for each pair.

$$1+20384=20385 2+10192=10194 4+5096=5100 7+2912=2919 8+2548=2556 13+1568=1581 14+1456=1470 16+1274=1290 26+784=810 28+728=756 32+637=669 49+416=465 52+392=444 56+364=420 91+224=315 98+208=306 104+196=300 112+182=294$$

The solution is the pair that gives sum 315.

$$a=91 b=224$$

Rewrite 104x^{2}+315x+196 as \left(104x^{2}+91x\right)+\left(224x+196\right).

$$\left(104x^{2}+91x\right)+\left(224x+196\right)$$

Factor out 13x in the first and 28 in the second group.

$$13x\left(8x+7\right)+28\left(8x+7\right)$$

Factor out common term 8x+7 by using distributive property.

$$\left(8x+7\right)\left(13x+28\right)$$

Rewrite the complete factored expression.

$$6\left(8x+7\right)\left(13x+28\right)$$

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.

$$624x^{2}+1890x+1176=0$$

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.

$$x=\frac{-1890±\sqrt{1890^{2}-4\times 624\times 1176}}{2\times 624}$$

Square 1890.

$$x=\frac{-1890±\sqrt{3572100-4\times 624\times 1176}}{2\times 624}$$

Multiply -4 times 624.

$$x=\frac{-1890±\sqrt{3572100-2496\times 1176}}{2\times 624}$$

Multiply -2496 times 1176.

$$x=\frac{-1890±\sqrt{3572100-2935296}}{2\times 624}$$

Add 3572100 to -2935296.

$$x=\frac{-1890±\sqrt{636804}}{2\times 624}$$

Take the square root of 636804.

$$x=\frac{-1890±798}{2\times 624}$$

Multiply 2 times 624.

$$x=\frac{-1890±798}{1248}$$

Now solve the equation x=\frac{-1890±798}{1248} when ± is plus. Add -1890 to 798.

$$x=-\frac{1092}{1248}$$

Reduce the fraction \frac{-1092}{1248} to lowest terms by extracting and canceling out 156.

$$x=-\frac{7}{8}$$

Now solve the equation x=\frac{-1890±798}{1248} when ± is minus. Subtract 798 from -1890.

$$x=-\frac{2688}{1248}$$

Reduce the fraction \frac{-2688}{1248} to lowest terms by extracting and canceling out 96.

$$x=-\frac{28}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{7}{8} for x_{1} and -\frac{28}{13} for x_{2}.

$$624x^{2}+1890x+1176=624\left(x-\left(-\frac{7}{8}\right)\right)\left(x-\left(-\frac{28}{13}\right)\right)$$

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

$$624x^{2}+1890x+1176=624\left(x+\frac{7}{8}\right)\left(x+\frac{28}{13}\right)$$

Add \frac{7}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$624x^{2}+1890x+1176=624\times \frac{8x+7}{8}\left(x+\frac{28}{13}\right)$$

Add \frac{28}{13} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$624x^{2}+1890x+1176=624\times \frac{8x+7}{8}\times \frac{13x+28}{13}$$

Multiply \frac{8x+7}{8} times \frac{13x+28}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$624x^{2}+1890x+1176=624\times \frac{\left(8x+7\right)\left(13x+28\right)}{8\times 13}$$

Multiply 8 times 13.

$$624x^{2}+1890x+1176=624\times \frac{\left(8x+7\right)\left(13x+28\right)}{104}$$

Cancel out 104, the greatest common factor in 624 and 104.

$$624x^{2}+1890x+1176=6\left(8x+7\right)\left(13x+28\right)$$

Evaluate

$$624x^{2}+1890x+1176$$