Solve 3x^2-34x+11

$$3 x ^ { 2 } - 34 x + 11$$

Factor

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

Solution Steps

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

$$a+b=-34 ab=3\times 11=33$$

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 -34.

$$a=-33 b=-1$$

Rewrite 3x^{2}-34x+11 as \left(3x^{2}-33x\right)+\left(-x+11\right).

$$\left(3x^{2}-33x\right)+\left(-x+11\right)$$

Factor out 3x in the first and -1 in the second group.

$$3x\left(x-11\right)-\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

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

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.

$$3x^{2}-34x+11=0$$

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.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 3\times 11}}{2\times 3}$$

Square -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 3\times 11}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-34\right)±\sqrt{1156-12\times 11}}{2\times 3}$$

Multiply -12 times 11.

$$x=\frac{-\left(-34\right)±\sqrt{1156-132}}{2\times 3}$$

Add 1156 to -132.

$$x=\frac{-\left(-34\right)±\sqrt{1024}}{2\times 3}$$

Take the square root of 1024.

$$x=\frac{-\left(-34\right)±32}{2\times 3}$$

The opposite of -34 is 34.

$$x=\frac{34±32}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{34±32}{6}$$

Now solve the equation x=\frac{34±32}{6} when ± is plus. Add 34 to 32.

$$x=\frac{66}{6}$$

Divide 66 by 6.

$$x=11$$

Now solve the equation x=\frac{34±32}{6} when ± is minus. Subtract 32 from 34.

$$x=\frac{2}{6}$$

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

$$x=\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11 for x_{1} and \frac{1}{3} for x_{2}.

$$3x^{2}-34x+11=3\left(x-11\right)\left(x-\frac{1}{3}\right)$$

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}-34x+11=3\left(x-11\right)\times \frac{3x-1}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}-34x+11=\left(x-11\right)\left(3x-1\right)$$

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 3

$$x ^ 2 -\frac{34}{3}x +\frac{11}{3} = 0$$

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

$$r + s = \frac{34}{3} rs = \frac{11}{3}$$

Two numbers r and s sum up to \frac{34}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{34}{3} = \frac{17}{3}. 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>

$$r = \frac{17}{3} - u s = \frac{17}{3} + u$$

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

$$(\frac{17}{3} - u) (\frac{17}{3} + u) = \frac{11}{3}$$

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

$$\frac{289}{9} - u^2 = \frac{11}{3}$$

Simplify the expression by subtracting \frac{289}{9} on both sides

$$-u^2 = \frac{11}{3}-\frac{289}{9} = -\frac{256}{9}$$

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

$$u^2 = \frac{256}{9} u = \pm\sqrt{\frac{256}{9}} = \pm \frac{16}{3}$$

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

$$r =\frac{17}{3} - \frac{16}{3} = 0.333 s = \frac{17}{3} + \frac{16}{3} = 11$$

Evaluate

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