Solve f(x)=4x^2-8x+3

$$f ( x ) = 4 x ^ { 2 } - 8 x + 3$$

Factor

$$\left(2x-3\right)\left(2x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-8 ab=4\times 3=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -8.

$$a=-6 b=-2$$

Rewrite 4x^{2}-8x+3 as \left(4x^{2}-6x\right)+\left(-2x+3\right).

$$\left(4x^{2}-6x\right)+\left(-2x+3\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(2x-3\right)-\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(2x-1\right)$$

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.

$$4x^{2}-8x+3=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\times 3}}{2\times 4}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-8\right)±\sqrt{64-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2\times 4}$$

Add 64 to -48.

$$x=\frac{-\left(-8\right)±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$x=\frac{-\left(-8\right)±4}{2\times 4}$$

The opposite of -8 is 8.

$$x=\frac{8±4}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{8±4}{8}$$

Now solve the equation x=\frac{8±4}{8} when ± is plus. Add 8 to 4.

$$x=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{8±4}{8} when ± is minus. Subtract 4 from 8.

$$x=\frac{4}{8}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{2} for x_{1} and \frac{1}{2} for x_{2}.

$$4x^{2}-8x+3=4\left(x-\frac{3}{2}\right)\left(x-\frac{1}{2}\right)$$

Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-8x+3=4\times \frac{2x-3}{2}\left(x-\frac{1}{2}\right)$$

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

$$4x^{2}-8x+3=4\times \frac{2x-3}{2}\times \frac{2x-1}{2}$$

Multiply \frac{2x-3}{2} times \frac{2x-1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-8x+3=4\times \frac{\left(2x-3\right)\left(2x-1\right)}{2\times 2}$$

Multiply 2 times 2.

$$4x^{2}-8x+3=4\times \frac{\left(2x-3\right)\left(2x-1\right)}{4}$$

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

$$4x^{2}-8x+3=\left(2x-3\right)\left(2x-1\right)$$

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

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

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

$$r + s = 2 rs = \frac{3}{4}$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{4}

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

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

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

Simplify the expression by subtracting 1 on both sides

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

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =1 - \frac{1}{2} = 0.500 s = 1 + \frac{1}{2} = 1.500$$

Evaluate

$$\left(2x-3\right)\left(2x-1\right)$$