Solve 7b^2+8b-12

$$7 b ^ { 2 } + 8 b - 12$$

Factor

$$\left(7b-6\right)\left(b+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7b^{2}+pb+qb-12. To find p and q, set up a system to be solved.

$$p+q=8 pq=7\left(-12\right)=-84$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 8.

$$p=-6 q=14$$

Rewrite 7b^{2}+8b-12 as \left(7b^{2}-6b\right)+\left(14b-12\right).

$$\left(7b^{2}-6b\right)+\left(14b-12\right)$$

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

$$b\left(7b-6\right)+2\left(7b-6\right)$$

Factor out common term 7b-6 by using distributive property.

$$\left(7b-6\right)\left(b+2\right)$$

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.

$$7b^{2}+8b-12=0$$

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.

$$b=\frac{-8±\sqrt{8^{2}-4\times 7\left(-12\right)}}{2\times 7}$$

Square 8.

$$b=\frac{-8±\sqrt{64-4\times 7\left(-12\right)}}{2\times 7}$$

Multiply -4 times 7.

$$b=\frac{-8±\sqrt{64-28\left(-12\right)}}{2\times 7}$$

Multiply -28 times -12.

$$b=\frac{-8±\sqrt{64+336}}{2\times 7}$$

Add 64 to 336.

$$b=\frac{-8±\sqrt{400}}{2\times 7}$$

Take the square root of 400.

$$b=\frac{-8±20}{2\times 7}$$

Multiply 2 times 7.

$$b=\frac{-8±20}{14}$$

Now solve the equation b=\frac{-8±20}{14} when ± is plus. Add -8 to 20.

$$b=\frac{12}{14}$$

Reduce the fraction \frac{12}{14} to lowest terms by extracting and canceling out 2.

$$b=\frac{6}{7}$$

Now solve the equation b=\frac{-8±20}{14} when ± is minus. Subtract 20 from -8.

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

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{6}{7} for x_{1} and -2 for x_{2}.

$$7b^{2}+8b-12=7\left(b-\frac{6}{7}\right)\left(b-\left(-2\right)\right)$$

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

$$7b^{2}+8b-12=7\left(b-\frac{6}{7}\right)\left(b+2\right)$$

Subtract \frac{6}{7} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7b^{2}+8b-12=7\times \frac{7b-6}{7}\left(b+2\right)$$

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

$$7b^{2}+8b-12=\left(7b-6\right)\left(b+2\right)$$

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

$$x ^ 2 +\frac{8}{7}x -\frac{12}{7} = 0$$

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

$$r + s = -\frac{8}{7} rs = -\frac{12}{7}$$

Two numbers r and s sum up to -\frac{8}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{7} = -\frac{4}{7}. 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>

$$r = -\frac{4}{7} - u s = -\frac{4}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{12}{7}

$$(-\frac{4}{7} - u) (-\frac{4}{7} + u) = -\frac{12}{7}$$

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

$$\frac{16}{49} - u^2 = -\frac{12}{7}$$

Simplify the expression by subtracting \frac{16}{49} on both sides

$$-u^2 = -\frac{12}{7}-\frac{16}{49} = -\frac{100}{49}$$

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

$$u^2 = \frac{100}{49} u = \pm\sqrt{\frac{100}{49}} = \pm \frac{10}{7}$$

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

$$r =-\frac{4}{7} - \frac{10}{7} = -2 s = -\frac{4}{7} + \frac{10}{7} = 0.857$$

Evaluate

$$\left(7b-6\right)\left(b+2\right)$$