Solve 8b^2-2b-3

$$8 b ^ { 2 } - 2 b - 3$$

Factor

$$\left(4b-3\right)\left(2b+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8b^{2}+pb+qb-3. To find p and q, set up a system to be solved.

$$p+q=-2 pq=8\left(-3\right)=-24$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$p=-6 q=4$$

Rewrite 8b^{2}-2b-3 as \left(8b^{2}-6b\right)+\left(4b-3\right).

$$\left(8b^{2}-6b\right)+\left(4b-3\right)$$

Factor out 2b in 8b^{2}-6b.

$$2b\left(4b-3\right)+4b-3$$

Factor out common term 4b-3 by using distributive property.

$$\left(4b-3\right)\left(2b+1\right)$$

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.

$$8b^{2}-2b-3=0$$

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.

$$b=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 8\left(-3\right)}}{2\times 8}$$

Square -2.

$$b=\frac{-\left(-2\right)±\sqrt{4-4\times 8\left(-3\right)}}{2\times 8}$$

Multiply -4 times 8.

$$b=\frac{-\left(-2\right)±\sqrt{4-32\left(-3\right)}}{2\times 8}$$

Multiply -32 times -3.

$$b=\frac{-\left(-2\right)±\sqrt{4+96}}{2\times 8}$$

Add 4 to 96.

$$b=\frac{-\left(-2\right)±\sqrt{100}}{2\times 8}$$

Take the square root of 100.

$$b=\frac{-\left(-2\right)±10}{2\times 8}$$

The opposite of -2 is 2.

$$b=\frac{2±10}{2\times 8}$$

Multiply 2 times 8.

$$b=\frac{2±10}{16}$$

Now solve the equation b=\frac{2±10}{16} when ± is plus. Add 2 to 10.

$$b=\frac{12}{16}$$

Reduce the fraction \frac{12}{16} to lowest terms by extracting and canceling out 4.

$$b=\frac{3}{4}$$

Now solve the equation b=\frac{2±10}{16} when ± is minus. Subtract 10 from 2.

$$b=-\frac{8}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

$$b=-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{4} for x_{1} and -\frac{1}{2} for x_{2}.

$$8b^{2}-2b-3=8\left(b-\frac{3}{4}\right)\left(b-\left(-\frac{1}{2}\right)\right)$$

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

$$8b^{2}-2b-3=8\left(b-\frac{3}{4}\right)\left(b+\frac{1}{2}\right)$$

Subtract \frac{3}{4} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8b^{2}-2b-3=8\times \frac{4b-3}{4}\left(b+\frac{1}{2}\right)$$

Add \frac{1}{2} to b by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8b^{2}-2b-3=8\times \frac{4b-3}{4}\times \frac{2b+1}{2}$$

Multiply \frac{4b-3}{4} times \frac{2b+1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8b^{2}-2b-3=8\times \frac{\left(4b-3\right)\left(2b+1\right)}{4\times 2}$$

Multiply 4 times 2.

$$8b^{2}-2b-3=8\times \frac{\left(4b-3\right)\left(2b+1\right)}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8b^{2}-2b-3=\left(4b-3\right)\left(2b+1\right)$$

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 8

$$x ^ 2 -\frac{1}{4}x -\frac{3}{8} = 0$$

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

$$r + s = \frac{1}{4} rs = -\frac{3}{8}$$

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

$$r = \frac{1}{8} - u s = \frac{1}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{8}

$$(\frac{1}{8} - u) (\frac{1}{8} + u) = -\frac{3}{8}$$

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

$$\frac{1}{64} - u^2 = -\frac{3}{8}$$

Simplify the expression by subtracting \frac{1}{64} on both sides

$$-u^2 = -\frac{3}{8}-\frac{1}{64} = -\frac{25}{64}$$

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

$$u^2 = \frac{25}{64} u = \pm\sqrt{\frac{25}{64}} = \pm \frac{5}{8}$$

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

$$r =\frac{1}{8} - \frac{5}{8} = -0.500 s = \frac{1}{8} + \frac{5}{8} = 0.750$$

Evaluate

$$\left(4b-3\right)\left(2b+1\right)$$