Solve 34y^2-66y-4

$$34 y ^ { 2 } - 66 y - 4$$

Factor

$$2\left(y-2\right)\left(17y+1\right)$$

Solution Steps

Factor out 2.

$$2\left(17y^{2}-33y-2\right)$$

Consider 17y^{2}-33y-2. Factor the expression by grouping. First, the expression needs to be rewritten as 17y^{2}+ay+by-2. To find a and b, set up a system to be solved.

$$a+b=-33 ab=17\left(-2\right)=-34$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -34.

$$1,-34 2,-17$$

Calculate the sum for each pair.

$$1-34=-33 2-17=-15$$

The solution is the pair that gives sum -33.

$$a=-34 b=1$$

Rewrite 17y^{2}-33y-2 as \left(17y^{2}-34y\right)+\left(y-2\right).

$$\left(17y^{2}-34y\right)+\left(y-2\right)$$

Factor out 17y in 17y^{2}-34y.

$$17y\left(y-2\right)+y-2$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(17y+1\right)$$

Rewrite the complete factored expression.

$$2\left(y-2\right)\left(17y+1\right)$$

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.

$$34y^{2}-66y-4=0$$

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.

$$y=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 34\left(-4\right)}}{2\times 34}$$

Square -66.

$$y=\frac{-\left(-66\right)±\sqrt{4356-4\times 34\left(-4\right)}}{2\times 34}$$

Multiply -4 times 34.

$$y=\frac{-\left(-66\right)±\sqrt{4356-136\left(-4\right)}}{2\times 34}$$

Multiply -136 times -4.

$$y=\frac{-\left(-66\right)±\sqrt{4356+544}}{2\times 34}$$

Add 4356 to 544.

$$y=\frac{-\left(-66\right)±\sqrt{4900}}{2\times 34}$$

Take the square root of 4900.

$$y=\frac{-\left(-66\right)±70}{2\times 34}$$

The opposite of -66 is 66.

$$y=\frac{66±70}{2\times 34}$$

Multiply 2 times 34.

$$y=\frac{66±70}{68}$$

Now solve the equation y=\frac{66±70}{68} when ± is plus. Add 66 to 70.

$$y=\frac{136}{68}$$

Divide 136 by 68.

$$y=2$$

Now solve the equation y=\frac{66±70}{68} when ± is minus. Subtract 70 from 66.

$$y=-\frac{4}{68}$$

Reduce the fraction \frac{-4}{68} to lowest terms by extracting and canceling out 4.

$$y=-\frac{1}{17}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{1}{17} for x_{2}.

$$34y^{2}-66y-4=34\left(y-2\right)\left(y-\left(-\frac{1}{17}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$34y^{2}-66y-4=34\left(y-2\right)\left(y+\frac{1}{17}\right)$$

Add \frac{1}{17} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$34y^{2}-66y-4=34\left(y-2\right)\times \frac{17y+1}{17}$$

Cancel out 17, the greatest common factor in 34 and 17.

$$34y^{2}-66y-4=2\left(y-2\right)\left(17y+1\right)$$

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 34

$$x ^ 2 -\frac{33}{17}x -\frac{2}{17} = 0$$

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

$$r + s = \frac{33}{17} rs = -\frac{2}{17}$$

Two numbers r and s sum up to \frac{33}{17} exactly when the average of the two numbers is \frac{1}{2}*\frac{33}{17} = \frac{33}{34}. 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>

$$r = \frac{33}{34} - u s = \frac{33}{34} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{17}

$$(\frac{33}{34} - u) (\frac{33}{34} + u) = -\frac{2}{17}$$

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

$$\frac{1089}{1156} - u^2 = -\frac{2}{17}$$

Simplify the expression by subtracting \frac{1089}{1156} on both sides

$$-u^2 = -\frac{2}{17}-\frac{1089}{1156} = -\frac{1225}{1156}$$

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

$$u^2 = \frac{1225}{1156} u = \pm\sqrt{\frac{1225}{1156}} = \pm \frac{35}{34}$$

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

$$r =\frac{33}{34} - \frac{35}{34} = -0.059 s = \frac{33}{34} + \frac{35}{34} = 2$$

Evaluate

$$2\left(y-2\right)\left(17y+1\right)$$