Solve 7+22x+3x^2

$$7 + 22 x + 3 x ^ { 2 }$$

Factor

$$\left(x+7\right)\left(3x+1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3x^{2}+22x+7$$

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=22 ab=3\times 7=21$$

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 21.

$$1,21 3,7$$

Calculate the sum for each pair.

$$1+21=22 3+7=10$$

The solution is the pair that gives sum 22.

$$a=1 b=21$$

Rewrite 3x^{2}+22x+7 as \left(3x^{2}+x\right)+\left(21x+7\right).

$$\left(3x^{2}+x\right)+\left(21x+7\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(3x+1\right)+7\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(x+7\right)$$

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.

$$3x^{2}+22x+7=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 3\times 7}}{2\times 3}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 3\times 7}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-22±\sqrt{484-12\times 7}}{2\times 3}$$

Multiply -12 times 7.

$$x=\frac{-22±\sqrt{484-84}}{2\times 3}$$

Add 484 to -84.

$$x=\frac{-22±\sqrt{400}}{2\times 3}$$

Take the square root of 400.

$$x=\frac{-22±20}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-22±20}{6}$$

Now solve the equation x=\frac{-22±20}{6} when ± is plus. Add -22 to 20.

$$x=-\frac{2}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-22±20}{6} when ± is minus. Subtract 20 from -22.

$$x=-\frac{42}{6}$$

Divide -42 by 6.

$$x=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{3} for x_{1} and -7 for x_{2}.

$$3x^{2}+22x+7=3\left(x-\left(-\frac{1}{3}\right)\right)\left(x-\left(-7\right)\right)$$

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

$$3x^{2}+22x+7=3\left(x+\frac{1}{3}\right)\left(x+7\right)$$

Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}+22x+7=3\times \frac{3x+1}{3}\left(x+7\right)$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}+22x+7=\left(3x+1\right)\left(x+7\right)$$

Evaluate

$$\left(x+7\right)\left(3x+1\right)$$