Solve 4b^2-21b+5

$$4 b ^ { 2 } - 21 b + 5$$

Factor

$$\left(b-5\right)\left(4b-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4b^{2}+pb+qb+5. To find p and q, set up a system to be solved.

$$p+q=-21 pq=4\times 5=20$$

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 20.

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

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -21.

$$p=-20 q=-1$$

Rewrite 4b^{2}-21b+5 as \left(4b^{2}-20b\right)+\left(-b+5\right).

$$\left(4b^{2}-20b\right)+\left(-b+5\right)$$

Factor out 4b in the first and -1 in the second group.

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

Factor out common term b-5 by using distributive property.

$$\left(b-5\right)\left(4b-1\right)$$

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.

$$4b^{2}-21b+5=0$$

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.

$$b=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\times 5}}{2\times 4}$$

Square -21.

$$b=\frac{-\left(-21\right)±\sqrt{441-4\times 4\times 5}}{2\times 4}$$

Multiply -4 times 4.

$$b=\frac{-\left(-21\right)±\sqrt{441-16\times 5}}{2\times 4}$$

Multiply -16 times 5.

$$b=\frac{-\left(-21\right)±\sqrt{441-80}}{2\times 4}$$

Add 441 to -80.

$$b=\frac{-\left(-21\right)±\sqrt{361}}{2\times 4}$$

Take the square root of 361.

$$b=\frac{-\left(-21\right)±19}{2\times 4}$$

The opposite of -21 is 21.

$$b=\frac{21±19}{2\times 4}$$

Multiply 2 times 4.

$$b=\frac{21±19}{8}$$

Now solve the equation b=\frac{21±19}{8} when ± is plus. Add 21 to 19.

$$b=\frac{40}{8}$$

Divide 40 by 8.

$$b=5$$

Now solve the equation b=\frac{21±19}{8} when ± is minus. Subtract 19 from 21.

$$b=\frac{2}{8}$$

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

$$b=\frac{1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and \frac{1}{4} for x_{2}.

$$4b^{2}-21b+5=4\left(b-5\right)\left(b-\frac{1}{4}\right)$$

Subtract \frac{1}{4} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4b^{2}-21b+5=4\left(b-5\right)\times \frac{4b-1}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4b^{2}-21b+5=\left(b-5\right)\left(4b-1\right)$$

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 4

$$x ^ 2 -\frac{21}{4}x +\frac{5}{4} = 0$$

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

$$r + s = \frac{21}{4} rs = \frac{5}{4}$$

Two numbers r and s sum up to \frac{21}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{21}{4} = \frac{21}{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>

$$r = \frac{21}{8} - u s = \frac{21}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{4}

$$(\frac{21}{8} - u) (\frac{21}{8} + u) = \frac{5}{4}$$

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

$$\frac{441}{64} - u^2 = \frac{5}{4}$$

Simplify the expression by subtracting \frac{441}{64} on both sides

$$-u^2 = \frac{5}{4}-\frac{441}{64} = -\frac{361}{64}$$

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

$$u^2 = \frac{361}{64} u = \pm\sqrt{\frac{361}{64}} = \pm \frac{19}{8}$$

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

$$r =\frac{21}{8} - \frac{19}{8} = 0.250 s = \frac{21}{8} + \frac{19}{8} = 5$$

Evaluate

$$\left(b-5\right)\left(4b-1\right)$$