Solve 11w^2-14w+3

$$11 w ^ { 2 } - 14 w + 3$$

Factor

$$\left(w-1\right)\left(11w-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11w^{2}+aw+bw+3. To find a and b, set up a system to be solved.

$$a+b=-14 ab=11\times 3=33$$

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 33.

$$-1,-33 -3,-11$$

Calculate the sum for each pair.

$$-1-33=-34 -3-11=-14$$

The solution is the pair that gives sum -14.

$$a=-11 b=-3$$

Rewrite 11w^{2}-14w+3 as \left(11w^{2}-11w\right)+\left(-3w+3\right).

$$\left(11w^{2}-11w\right)+\left(-3w+3\right)$$

Factor out 11w in the first and -3 in the second group.

$$11w\left(w-1\right)-3\left(w-1\right)$$

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

$$\left(w-1\right)\left(11w-3\right)$$

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.

$$11w^{2}-14w+3=0$$

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.

$$w=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 11\times 3}}{2\times 11}$$

Square -14.

$$w=\frac{-\left(-14\right)±\sqrt{196-4\times 11\times 3}}{2\times 11}$$

Multiply -4 times 11.

$$w=\frac{-\left(-14\right)±\sqrt{196-44\times 3}}{2\times 11}$$

Multiply -44 times 3.

$$w=\frac{-\left(-14\right)±\sqrt{196-132}}{2\times 11}$$

Add 196 to -132.

$$w=\frac{-\left(-14\right)±\sqrt{64}}{2\times 11}$$

Take the square root of 64.

$$w=\frac{-\left(-14\right)±8}{2\times 11}$$

The opposite of -14 is 14.

$$w=\frac{14±8}{2\times 11}$$

Multiply 2 times 11.

$$w=\frac{14±8}{22}$$

Now solve the equation w=\frac{14±8}{22} when ± is plus. Add 14 to 8.

$$w=\frac{22}{22}$$

Divide 22 by 22.

$$w=1$$

Now solve the equation w=\frac{14±8}{22} when ± is minus. Subtract 8 from 14.

$$w=\frac{6}{22}$$

Reduce the fraction \frac{6}{22} to lowest terms by extracting and canceling out 2.

$$w=\frac{3}{11}$$

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{3}{11} for x_{2}.

$$11w^{2}-14w+3=11\left(w-1\right)\left(w-\frac{3}{11}\right)$$

Subtract \frac{3}{11} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$11w^{2}-14w+3=11\left(w-1\right)\times \frac{11w-3}{11}$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11w^{2}-14w+3=\left(w-1\right)\left(11w-3\right)$$

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 11

$$x ^ 2 -\frac{14}{11}x +\frac{3}{11} = 0$$

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

$$r + s = \frac{14}{11} rs = \frac{3}{11}$$

Two numbers r and s sum up to \frac{14}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{14}{11} = \frac{7}{11}. 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>

$$r = \frac{7}{11} - u s = \frac{7}{11} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{11}

$$(\frac{7}{11} - u) (\frac{7}{11} + u) = \frac{3}{11}$$

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

$$\frac{49}{121} - u^2 = \frac{3}{11}$$

Simplify the expression by subtracting \frac{49}{121} on both sides

$$-u^2 = \frac{3}{11}-\frac{49}{121} = -\frac{16}{121}$$

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

$$u^2 = \frac{16}{121} u = \pm\sqrt{\frac{16}{121}} = \pm \frac{4}{11}$$

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

$$r =\frac{7}{11} - \frac{4}{11} = 0.273 s = \frac{7}{11} + \frac{4}{11} = 1$$

Evaluate

$$\left(w-1\right)\left(11w-3\right)$$