Solve 20x^2-x-1

$$20 x ^ { 2 } - x - 1$$

Factor

$$\left(4x-1\right)\left(5x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 20x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-1 ab=20\left(-1\right)=-20$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -20.

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

Calculate the sum for each pair.

$$1-20=-19 2-10=-8 4-5=-1$$

The solution is the pair that gives sum -1.

$$a=-5 b=4$$

Rewrite 20x^{2}-x-1 as \left(20x^{2}-5x\right)+\left(4x-1\right).

$$\left(20x^{2}-5x\right)+\left(4x-1\right)$$

Factor out 5x in 20x^{2}-5x.

$$5x\left(4x-1\right)+4x-1$$

Factor out common term 4x-1 by using distributive property.

$$\left(4x-1\right)\left(5x+1\right)$$

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.

$$20x^{2}-x-1=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 20\left(-1\right)}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-1\right)±\sqrt{1-80\left(-1\right)}}{2\times 20}$$

Multiply -80 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+80}}{2\times 20}$$

Add 1 to 80.

$$x=\frac{-\left(-1\right)±\sqrt{81}}{2\times 20}$$

Take the square root of 81.

$$x=\frac{-\left(-1\right)±9}{2\times 20}$$

The opposite of -1 is 1.

$$x=\frac{1±9}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{1±9}{40}$$

Now solve the equation x=\frac{1±9}{40} when ± is plus. Add 1 to 9.

$$x=\frac{10}{40}$$

Reduce the fraction \frac{10}{40} to lowest terms by extracting and canceling out 10.

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

Now solve the equation x=\frac{1±9}{40} when ± is minus. Subtract 9 from 1.

$$x=-\frac{8}{40}$$

Reduce the fraction \frac{-8}{40} to lowest terms by extracting and canceling out 8.

$$x=-\frac{1}{5}$$

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

$$20x^{2}-x-1=20\left(x-\frac{1}{4}\right)\left(x-\left(-\frac{1}{5}\right)\right)$$

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

$$20x^{2}-x-1=20\left(x-\frac{1}{4}\right)\left(x+\frac{1}{5}\right)$$

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

$$20x^{2}-x-1=20\times \frac{4x-1}{4}\left(x+\frac{1}{5}\right)$$

Add \frac{1}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$20x^{2}-x-1=20\times \frac{4x-1}{4}\times \frac{5x+1}{5}$$

Multiply \frac{4x-1}{4} times \frac{5x+1}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$20x^{2}-x-1=20\times \frac{\left(4x-1\right)\left(5x+1\right)}{4\times 5}$$

Multiply 4 times 5.

$$20x^{2}-x-1=20\times \frac{\left(4x-1\right)\left(5x+1\right)}{20}$$

Cancel out 20, the greatest common factor in 20 and 20.

$$20x^{2}-x-1=\left(4x-1\right)\left(5x+1\right)$$

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 20

$$x ^ 2 -\frac{1}{20}x -\frac{1}{20} = 0$$

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

$$r + s = \frac{1}{20} rs = -\frac{1}{20}$$

Two numbers r and s sum up to \frac{1}{20} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{20} = \frac{1}{40}. 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>

$$r = \frac{1}{40} - u s = \frac{1}{40} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{20}

$$(\frac{1}{40} - u) (\frac{1}{40} + u) = -\frac{1}{20}$$

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

$$\frac{1}{1600} - u^2 = -\frac{1}{20}$$

Simplify the expression by subtracting \frac{1}{1600} on both sides

$$-u^2 = -\frac{1}{20}-\frac{1}{1600} = -\frac{81}{1600}$$

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

$$u^2 = \frac{81}{1600} u = \pm\sqrt{\frac{81}{1600}} = \pm \frac{9}{40}$$

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

$$r =\frac{1}{40} - \frac{9}{40} = -0.200 s = \frac{1}{40} + \frac{9}{40} = 0.250$$

Evaluate

$$\left(4x-1\right)\left(5x+1\right)$$