Solve 7b^2+15b+2

$$7 b ^ { 2 } + 15 b + 2$$

Factor

$$\left(b+2\right)\left(7b+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7b^{2}+pb+qb+2. To find p and q, set up a system to be solved.

$$p+q=15 pq=7\times 2=14$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 15.

$$p=1 q=14$$

Rewrite 7b^{2}+15b+2 as \left(7b^{2}+b\right)+\left(14b+2\right).

$$\left(7b^{2}+b\right)+\left(14b+2\right)$$

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

$$b\left(7b+1\right)+2\left(7b+1\right)$$

Factor out common term 7b+1 by using distributive property.

$$\left(7b+1\right)\left(b+2\right)$$

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.

$$7b^{2}+15b+2=0$$

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.

$$b=\frac{-15±\sqrt{15^{2}-4\times 7\times 2}}{2\times 7}$$

Square 15.

$$b=\frac{-15±\sqrt{225-4\times 7\times 2}}{2\times 7}$$

Multiply -4 times 7.

$$b=\frac{-15±\sqrt{225-28\times 2}}{2\times 7}$$

Multiply -28 times 2.

$$b=\frac{-15±\sqrt{225-56}}{2\times 7}$$

Add 225 to -56.

$$b=\frac{-15±\sqrt{169}}{2\times 7}$$

Take the square root of 169.

$$b=\frac{-15±13}{2\times 7}$$

Multiply 2 times 7.

$$b=\frac{-15±13}{14}$$

Now solve the equation b=\frac{-15±13}{14} when ± is plus. Add -15 to 13.

$$b=-\frac{2}{14}$$

Reduce the fraction \frac{-2}{14} to lowest terms by extracting and canceling out 2.

$$b=-\frac{1}{7}$$

Now solve the equation b=\frac{-15±13}{14} when ± is minus. Subtract 13 from -15.

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

Divide -28 by 14.

$$b=-2$$

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

$$7b^{2}+15b+2=7\left(b-\left(-\frac{1}{7}\right)\right)\left(b-\left(-2\right)\right)$$

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

$$7b^{2}+15b+2=7\left(b+\frac{1}{7}\right)\left(b+2\right)$$

Add \frac{1}{7} to b by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7b^{2}+15b+2=7\times \frac{7b+1}{7}\left(b+2\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7b^{2}+15b+2=\left(7b+1\right)\left(b+2\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 7

$$x ^ 2 +\frac{15}{7}x +\frac{2}{7} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{15}{7} rs = \frac{2}{7}$$

Two numbers r and s sum up to -\frac{15}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{15}{7} = -\frac{15}{14}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{15}{14} - u s = -\frac{15}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2}{7}

$$(-\frac{15}{14} - u) (-\frac{15}{14} + u) = \frac{2}{7}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{225}{196} - u^2 = \frac{2}{7}$$

Simplify the expression by subtracting \frac{225}{196} on both sides

$$-u^2 = \frac{2}{7}-\frac{225}{196} = -\frac{169}{196}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{169}{196} u = \pm\sqrt{\frac{169}{196}} = \pm \frac{13}{14}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{15}{14} - \frac{13}{14} = -2 s = -\frac{15}{14} + \frac{13}{14} = -0.143$$

Evaluate

$$\left(b+2\right)\left(7b+1\right)$$