Solve 25x^2+58x+16

$$25 { x }^{ 2 } +58x+16$$

Factor

$$\left(x+2\right)\left(25x+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=58 ab=25\times 16=400$$

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 400.

$$1,400 2,200 4,100 5,80 8,50 10,40 16,25 20,20$$

Calculate the sum for each pair.

$$1+400=401 2+200=202 4+100=104 5+80=85 8+50=58 10+40=50 16+25=41 20+20=40$$

The solution is the pair that gives sum 58.

$$a=8 b=50$$

Rewrite 25x^{2}+58x+16 as \left(25x^{2}+8x\right)+\left(50x+16\right).

$$\left(25x^{2}+8x\right)+\left(50x+16\right)$$

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

$$x\left(25x+8\right)+2\left(25x+8\right)$$

Factor out common term 25x+8 by using distributive property.

$$\left(25x+8\right)\left(x+2\right)$$

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.

$$25x^{2}+58x+16=0$$

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.

$$x=\frac{-58±\sqrt{58^{2}-4\times 25\times 16}}{2\times 25}$$

Square 58.

$$x=\frac{-58±\sqrt{3364-4\times 25\times 16}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-58±\sqrt{3364-100\times 16}}{2\times 25}$$

Multiply -100 times 16.

$$x=\frac{-58±\sqrt{3364-1600}}{2\times 25}$$

Add 3364 to -1600.

$$x=\frac{-58±\sqrt{1764}}{2\times 25}$$

Take the square root of 1764.

$$x=\frac{-58±42}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-58±42}{50}$$

Now solve the equation x=\frac{-58±42}{50} when ± is plus. Add -58 to 42.

$$x=-\frac{16}{50}$$

Reduce the fraction \frac{-16}{50} to lowest terms by extracting and canceling out 2.

$$x=-\frac{8}{25}$$

Now solve the equation x=\frac{-58±42}{50} when ± is minus. Subtract 42 from -58.

$$x=-\frac{100}{50}$$

Divide -100 by 50.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{8}{25} for x_{1} and -2 for x_{2}.

$$25x^{2}+58x+16=25\left(x-\left(-\frac{8}{25}\right)\right)\left(x-\left(-2\right)\right)$$

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

$$25x^{2}+58x+16=25\left(x+\frac{8}{25}\right)\left(x+2\right)$$

Add \frac{8}{25} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$25x^{2}+58x+16=25\times \frac{25x+8}{25}\left(x+2\right)$$

Cancel out 25, the greatest common factor in 25 and 25.

$$25x^{2}+58x+16=\left(25x+8\right)\left(x+2\right)$$

Evaluate

$$\left(x+2\right)\left(25x+8\right)$$