Solve 7x^2+35x+42

$$7 { x }^{ 2 } +35x+42$$

Factor

$$7\left(x+2\right)\left(x+3\right)$$

Solution Steps

Factor out 7.

$$7\left(x^{2}+5x+6\right)$$

Consider x^{2}+5x+6. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\times 6=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=2 b=3$$

Rewrite x^{2}+5x+6 as \left(x^{2}+2x\right)+\left(3x+6\right).

$$\left(x^{2}+2x\right)+\left(3x+6\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x+2\right)+3\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$7\left(x+2\right)\left(x+3\right)$$

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.

$$7x^{2}+35x+42=0$$

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.

$$x=\frac{-35±\sqrt{35^{2}-4\times 7\times 42}}{2\times 7}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\times 7\times 42}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-35±\sqrt{1225-28\times 42}}{2\times 7}$$

Multiply -28 times 42.

$$x=\frac{-35±\sqrt{1225-1176}}{2\times 7}$$

Add 1225 to -1176.

$$x=\frac{-35±\sqrt{49}}{2\times 7}$$

Take the square root of 49.

$$x=\frac{-35±7}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-35±7}{14}$$

Now solve the equation x=\frac{-35±7}{14} when ± is plus. Add -35 to 7.

$$x=-\frac{28}{14}$$

Divide -28 by 14.

$$x=-2$$

Now solve the equation x=\frac{-35±7}{14} when ± is minus. Subtract 7 from -35.

$$x=-\frac{42}{14}$$

Divide -42 by 14.

$$x=-3$$

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

$$7x^{2}+35x+42=7\left(x-\left(-2\right)\right)\left(x-\left(-3\right)\right)$$

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

$$7x^{2}+35x+42=7\left(x+2\right)\left(x+3\right)$$

Evaluate

$$7\left(x+2\right)\left(x+3\right)$$