Solve -4a^2+4a-1

$$- 4 a ^ { 2 } + 4 a - 1$$

Factor

$$-\left(2a-1\right)^{2}$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -4a^{2}+pa+qa-1. To find p and q, set up a system to be solved.

$$p+q=4 pq=-4\left(-1\right)=4$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 4.

$$p=2 q=2$$

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

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

Factor out -2a in -4a^{2}+2a.

$$-2a\left(2a-1\right)+2a-1$$

Factor out common term 2a-1 by using distributive property.

$$\left(2a-1\right)\left(-2a+1\right)$$

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.

$$-4a^{2}+4a-1=0$$

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.

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

Square 4.

$$a=\frac{-4±\sqrt{16-4\left(-4\right)\left(-1\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-4±\sqrt{16+16\left(-1\right)}}{2\left(-4\right)}$$

Multiply 16 times -1.

$$a=\frac{-4±\sqrt{16-16}}{2\left(-4\right)}$$

Add 16 to -16.

$$a=\frac{-4±\sqrt{0}}{2\left(-4\right)}$$

Take the square root of 0.

$$a=\frac{-4±0}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{-4±0}{-8}$$

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

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

Subtract \frac{1}{2} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Subtract \frac{1}{2} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-4a^{2}+4a-1=-4\times \frac{-2a+1}{-2}\times \frac{-2a+1}{-2}$$

Multiply \frac{-2a+1}{-2} times \frac{-2a+1}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

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

Multiply -2 times -2.

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

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

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

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.

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

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

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

Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{2}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

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

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = \frac{1}{4}$$

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

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

Simplify the expression by subtracting \frac{1}{4} on both sides

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

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

$$u^2 = 0 u = 0$$

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

$$r = s = \frac{1}{2} = 0.500$$

Evaluate

$$-\left(2a-1\right)^{2}$$