Solve 4x^2-3x-1

$$4 x ^ { 2 } - 3 x - 1$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-1\right)=-4$$

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 -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

Rewrite 4x^{2}-3x-1 as \left(4x^{2}-4x\right)+\left(x-1\right).

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

Factor out 4x in 4x^{2}-4x.

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

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

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

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.

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

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 4\left(-1\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-3\right)±\sqrt{9-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+16}}{2\times 4}$$

Add 9 to 16.

$$x=\frac{-\left(-3\right)±\sqrt{25}}{2\times 4}$$

Take the square root of 25.

$$x=\frac{-\left(-3\right)±5}{2\times 4}$$

The opposite of -3 is 3.

$$x=\frac{3±5}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{3±5}{8}$$

Now solve the equation x=\frac{3±5}{8} when ± is plus. Add 3 to 5.

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

Divide 8 by 8.

$$x=1$$

Now solve the equation x=\frac{3±5}{8} when ± is minus. Subtract 5 from 3.

$$x=-\frac{2}{8}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Two numbers r and s sum up to \frac{3}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{4} = \frac{3}{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>

$$r = \frac{3}{8} - u s = \frac{3}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{4}

$$(\frac{3}{8} - u) (\frac{3}{8} + u) = -\frac{1}{4}$$

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

$$\frac{9}{64} - u^2 = -\frac{1}{4}$$

Simplify the expression by subtracting \frac{9}{64} on both sides

$$-u^2 = -\frac{1}{4}-\frac{9}{64} = -\frac{25}{64}$$

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.

$$r =\frac{3}{8} - \frac{5}{8} = -0.250 s = \frac{3}{8} + \frac{5}{8} = 1$$

Evaluate

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