Solve 2x^2+35x+125

$$2 { x }^{ 2 } +35x+125$$

Factor

$$\left(x+5\right)\left(2x+25\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+125. To find a and b, set up a system to be solved.

$$a+b=35 ab=2\times 125=250$$

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 250.

$$1,250 2,125 5,50 10,25$$

Calculate the sum for each pair.

$$1+250=251 2+125=127 5+50=55 10+25=35$$

The solution is the pair that gives sum 35.

$$a=10 b=25$$

Rewrite 2x^{2}+35x+125 as \left(2x^{2}+10x\right)+\left(25x+125\right).

$$\left(2x^{2}+10x\right)+\left(25x+125\right)$$

Factor out 2x in the first and 25 in the second group.

$$2x\left(x+5\right)+25\left(x+5\right)$$

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

$$\left(x+5\right)\left(2x+25\right)$$

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.

$$2x^{2}+35x+125=0$$

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.

$$x=\frac{-35±\sqrt{35^{2}-4\times 2\times 125}}{2\times 2}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\times 2\times 125}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-35±\sqrt{1225-8\times 125}}{2\times 2}$$

Multiply -8 times 125.

$$x=\frac{-35±\sqrt{1225-1000}}{2\times 2}$$

Add 1225 to -1000.

$$x=\frac{-35±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-35±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-35±15}{4}$$

Now solve the equation x=\frac{-35±15}{4} when ± is plus. Add -35 to 15.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

Now solve the equation x=\frac{-35±15}{4} when ± is minus. Subtract 15 from -35.

$$x=-\frac{50}{4}$$

Reduce the fraction \frac{-50}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{25}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -\frac{25}{2} for x_{2}.

$$2x^{2}+35x+125=2\left(x-\left(-5\right)\right)\left(x-\left(-\frac{25}{2}\right)\right)$$

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

$$2x^{2}+35x+125=2\left(x+5\right)\left(x+\frac{25}{2}\right)$$

Add \frac{25}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}+35x+125=2\left(x+5\right)\times \frac{2x+25}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2x^{2}+35x+125=\left(x+5\right)\left(2x+25\right)$$

Evaluate

$$\left(x+5\right)\left(2x+25\right)$$