Solve 16x^2+76x+90

$$16 { x }^{ 2 } +76x+90$$

Factor

$$2\left(2x+5\right)\left(4x+9\right)$$

Solution Steps

Factor out 2.

$$2\left(8x^{2}+38x+45\right)$$

Consider 8x^{2}+38x+45. Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+45. To find a and b, set up a system to be solved.

$$a+b=38 ab=8\times 45=360$$

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 360.

$$1,360 2,180 3,120 4,90 5,72 6,60 8,45 9,40 10,36 12,30 15,24 18,20$$

Calculate the sum for each pair.

$$1+360=361 2+180=182 3+120=123 4+90=94 5+72=77 6+60=66 8+45=53 9+40=49 10+36=46 12+30=42 15+24=39 18+20=38$$

The solution is the pair that gives sum 38.

$$a=18 b=20$$

Rewrite 8x^{2}+38x+45 as \left(8x^{2}+18x\right)+\left(20x+45\right).

$$\left(8x^{2}+18x\right)+\left(20x+45\right)$$

Factor out 2x in the first and 5 in the second group.

$$2x\left(4x+9\right)+5\left(4x+9\right)$$

Factor out common term 4x+9 by using distributive property.

$$\left(4x+9\right)\left(2x+5\right)$$

Rewrite the complete factored expression.

$$2\left(4x+9\right)\left(2x+5\right)$$

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.

$$16x^{2}+76x+90=0$$

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.

$$x=\frac{-76±\sqrt{76^{2}-4\times 16\times 90}}{2\times 16}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\times 16\times 90}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-76±\sqrt{5776-64\times 90}}{2\times 16}$$

Multiply -64 times 90.

$$x=\frac{-76±\sqrt{5776-5760}}{2\times 16}$$

Add 5776 to -5760.

$$x=\frac{-76±\sqrt{16}}{2\times 16}$$

Take the square root of 16.

$$x=\frac{-76±4}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-76±4}{32}$$

Now solve the equation x=\frac{-76±4}{32} when ± is plus. Add -76 to 4.

$$x=-\frac{72}{32}$$

Reduce the fraction \frac{-72}{32} to lowest terms by extracting and canceling out 8.

$$x=-\frac{9}{4}$$

Now solve the equation x=\frac{-76±4}{32} when ± is minus. Subtract 4 from -76.

$$x=-\frac{80}{32}$$

Reduce the fraction \frac{-80}{32} to lowest terms by extracting and canceling out 16.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{9}{4} for x_{1} and -\frac{5}{2} for x_{2}.

$$16x^{2}+76x+90=16\left(x-\left(-\frac{9}{4}\right)\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$16x^{2}+76x+90=16\left(x+\frac{9}{4}\right)\left(x+\frac{5}{2}\right)$$

Add \frac{9}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+76x+90=16\times \frac{4x+9}{4}\left(x+\frac{5}{2}\right)$$

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

$$16x^{2}+76x+90=16\times \frac{4x+9}{4}\times \frac{2x+5}{2}$$

Multiply \frac{4x+9}{4} times \frac{2x+5}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+76x+90=16\times \frac{\left(4x+9\right)\left(2x+5\right)}{4\times 2}$$

Multiply 4 times 2.

$$16x^{2}+76x+90=16\times \frac{\left(4x+9\right)\left(2x+5\right)}{8}$$

Cancel out 8, the greatest common factor in 16 and 8.

$$16x^{2}+76x+90=2\left(4x+9\right)\left(2x+5\right)$$

Evaluate

$$16x^{2}+76x+90$$