Solve 5b^2-14b+8

$$5 b ^ { 2 } - 14 b + 8$$

Factor

$$\left(b-2\right)\left(5b-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5b^{2}+pb+qb+8. To find p and q, set up a system to be solved.

$$p+q=-14 pq=5\times 8=40$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Calculate the sum for each pair.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

The solution is the pair that gives sum -14.

$$p=-10 q=-4$$

Rewrite 5b^{2}-14b+8 as \left(5b^{2}-10b\right)+\left(-4b+8\right).

$$\left(5b^{2}-10b\right)+\left(-4b+8\right)$$

Factor out 5b in the first and -4 in the second group.

$$5b\left(b-2\right)-4\left(b-2\right)$$

Factor out common term b-2 by using distributive property.

$$\left(b-2\right)\left(5b-4\right)$$

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.

$$5b^{2}-14b+8=0$$

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.

$$b=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\times 8}}{2\times 5}$$

Square -14.

$$b=\frac{-\left(-14\right)±\sqrt{196-4\times 5\times 8}}{2\times 5}$$

Multiply -4 times 5.

$$b=\frac{-\left(-14\right)±\sqrt{196-20\times 8}}{2\times 5}$$

Multiply -20 times 8.

$$b=\frac{-\left(-14\right)±\sqrt{196-160}}{2\times 5}$$

Add 196 to -160.

$$b=\frac{-\left(-14\right)±\sqrt{36}}{2\times 5}$$

Take the square root of 36.

$$b=\frac{-\left(-14\right)±6}{2\times 5}$$

The opposite of -14 is 14.

$$b=\frac{14±6}{2\times 5}$$

Multiply 2 times 5.

$$b=\frac{14±6}{10}$$

Now solve the equation b=\frac{14±6}{10} when ± is plus. Add 14 to 6.

$$b=\frac{20}{10}$$

Divide 20 by 10.

$$b=2$$

Now solve the equation b=\frac{14±6}{10} when ± is minus. Subtract 6 from 14.

$$b=\frac{8}{10}$$

Reduce the fraction \frac{8}{10} to lowest terms by extracting and canceling out 2.

$$b=\frac{4}{5}$$

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 \frac{4}{5} for x_{2}.

$$5b^{2}-14b+8=5\left(b-2\right)\left(b-\frac{4}{5}\right)$$

Subtract \frac{4}{5} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5b^{2}-14b+8=5\left(b-2\right)\times \frac{5b-4}{5}$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5b^{2}-14b+8=\left(b-2\right)\left(5b-4\right)$$

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 5

$$x ^ 2 -\frac{14}{5}x +\frac{8}{5} = 0$$

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

$$r + s = \frac{14}{5} rs = \frac{8}{5}$$

Two numbers r and s sum up to \frac{14}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{14}{5} = \frac{7}{5}. 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>

$$r = \frac{7}{5} - u s = \frac{7}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{5}

$$(\frac{7}{5} - u) (\frac{7}{5} + u) = \frac{8}{5}$$

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

$$\frac{49}{25} - u^2 = \frac{8}{5}$$

Simplify the expression by subtracting \frac{49}{25} on both sides

$$-u^2 = \frac{8}{5}-\frac{49}{25} = -\frac{9}{25}$$

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

$$u^2 = \frac{9}{25} u = \pm\sqrt{\frac{9}{25}} = \pm \frac{3}{5}$$

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

$$r =\frac{7}{5} - \frac{3}{5} = 0.800 s = \frac{7}{5} + \frac{3}{5} = 2.000$$

Evaluate

$$\left(b-2\right)\left(5b-4\right)$$