Solve 33x^2-67x+20

$$33 x ^ { 2 } - 67 x + 20$$

Factor

$$\left(3x-5\right)\left(11x-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 33x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=-67 ab=33\times 20=660$$

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 660.

$$-1,-660 -2,-330 -3,-220 -4,-165 -5,-132 -6,-110 -10,-66 -11,-60 -12,-55 -15,-44 -20,-33 -22,-30$$

Calculate the sum for each pair.

$$-1-660=-661 -2-330=-332 -3-220=-223 -4-165=-169 -5-132=-137 -6-110=-116 -10-66=-76 -11-60=-71 -12-55=-67 -15-44=-59 -20-33=-53 -22-30=-52$$

The solution is the pair that gives sum -67.

$$a=-55 b=-12$$

Rewrite 33x^{2}-67x+20 as \left(33x^{2}-55x\right)+\left(-12x+20\right).

$$\left(33x^{2}-55x\right)+\left(-12x+20\right)$$

Factor out 11x in the first and -4 in the second group.

$$11x\left(3x-5\right)-4\left(3x-5\right)$$

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(11x-4\right)$$

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.

$$33x^{2}-67x+20=0$$

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.

$$x=\frac{-\left(-67\right)±\sqrt{\left(-67\right)^{2}-4\times 33\times 20}}{2\times 33}$$

Square -67.

$$x=\frac{-\left(-67\right)±\sqrt{4489-4\times 33\times 20}}{2\times 33}$$

Multiply -4 times 33.

$$x=\frac{-\left(-67\right)±\sqrt{4489-132\times 20}}{2\times 33}$$

Multiply -132 times 20.

$$x=\frac{-\left(-67\right)±\sqrt{4489-2640}}{2\times 33}$$

Add 4489 to -2640.

$$x=\frac{-\left(-67\right)±\sqrt{1849}}{2\times 33}$$

Take the square root of 1849.

$$x=\frac{-\left(-67\right)±43}{2\times 33}$$

The opposite of -67 is 67.

$$x=\frac{67±43}{2\times 33}$$

Multiply 2 times 33.

$$x=\frac{67±43}{66}$$

Now solve the equation x=\frac{67±43}{66} when ± is plus. Add 67 to 43.

$$x=\frac{110}{66}$$

Reduce the fraction \frac{110}{66} to lowest terms by extracting and canceling out 22.

$$x=\frac{5}{3}$$

Now solve the equation x=\frac{67±43}{66} when ± is minus. Subtract 43 from 67.

$$x=\frac{24}{66}$$

Reduce the fraction \frac{24}{66} to lowest terms by extracting and canceling out 6.

$$x=\frac{4}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{3} for x_{1} and \frac{4}{11} for x_{2}.

$$33x^{2}-67x+20=33\left(x-\frac{5}{3}\right)\left(x-\frac{4}{11}\right)$$

Subtract \frac{5}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$33x^{2}-67x+20=33\times \frac{3x-5}{3}\left(x-\frac{4}{11}\right)$$

Subtract \frac{4}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$33x^{2}-67x+20=33\times \frac{3x-5}{3}\times \frac{11x-4}{11}$$

Multiply \frac{3x-5}{3} times \frac{11x-4}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$33x^{2}-67x+20=33\times \frac{\left(3x-5\right)\left(11x-4\right)}{3\times 11}$$

Multiply 3 times 11.

$$33x^{2}-67x+20=33\times \frac{\left(3x-5\right)\left(11x-4\right)}{33}$$

Cancel out 33, the greatest common factor in 33 and 33.

$$33x^{2}-67x+20=\left(3x-5\right)\left(11x-4\right)$$

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 33

$$x ^ 2 -\frac{67}{33}x +\frac{20}{33} = 0$$

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

$$r + s = \frac{67}{33} rs = \frac{20}{33}$$

Two numbers r and s sum up to \frac{67}{33} exactly when the average of the two numbers is \frac{1}{2}*\frac{67}{33} = \frac{67}{66}. 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>

$$r = \frac{67}{66} - u s = \frac{67}{66} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{20}{33}

$$(\frac{67}{66} - u) (\frac{67}{66} + u) = \frac{20}{33}$$

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

$$\frac{4489}{4356} - u^2 = \frac{20}{33}$$

Simplify the expression by subtracting \frac{4489}{4356} on both sides

$$-u^2 = \frac{20}{33}-\frac{4489}{4356} = -\frac{1849}{4356}$$

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

$$u^2 = \frac{1849}{4356} u = \pm\sqrt{\frac{1849}{4356}} = \pm \frac{43}{66}$$

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

$$r =\frac{67}{66} - \frac{43}{66} = 0.364 s = \frac{67}{66} + \frac{43}{66} = 1.667$$

Evaluate

$$\left(3x-5\right)\left(11x-4\right)$$