Solve -4x^2-3x+1

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

Factor

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

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.

$$a+b=-3 ab=-4=-4$$

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.

$$a=1 b=-4$$

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

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

Factor out -x in the first and -1 in the second group.

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

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

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

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.

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

Square -3.

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

Multiply -4 times -4.

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

Add 9 to 16.

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

Take the square root of 25.

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

The opposite of -3 is 3.

$$x=\frac{3±5}{2\left(-4\right)}$$

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}.

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

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

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

Subtract \frac{1}{4} from x by finding a common denominator and subtracting 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.

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

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.

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

Evaluate

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