Solve 2x^2-39x+70

$$2 x ^ { 2 } - 39 x + 70$$

Factor

$$\left(x-2\right)\left(2x-35\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+70. To find a and b, set up a system to be solved.

$$a+b=-39 ab=2\times 70=140$$

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 140.

$$-1,-140 -2,-70 -4,-35 -5,-28 -7,-20 -10,-14$$

Calculate the sum for each pair.

$$-1-140=-141 -2-70=-72 -4-35=-39 -5-28=-33 -7-20=-27 -10-14=-24$$

The solution is the pair that gives sum -39.

$$a=-35 b=-4$$

Rewrite 2x^{2}-39x+70 as \left(2x^{2}-35x\right)+\left(-4x+70\right).

$$\left(2x^{2}-35x\right)+\left(-4x+70\right)$$

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

$$x\left(2x-35\right)-2\left(2x-35\right)$$

Factor out common term 2x-35 by using distributive property.

$$\left(2x-35\right)\left(x-2\right)$$

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.

$$2x^{2}-39x+70=0$$

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.

$$x=\frac{-\left(-39\right)±\sqrt{\left(-39\right)^{2}-4\times 2\times 70}}{2\times 2}$$

Square -39.

$$x=\frac{-\left(-39\right)±\sqrt{1521-4\times 2\times 70}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-39\right)±\sqrt{1521-8\times 70}}{2\times 2}$$

Multiply -8 times 70.

$$x=\frac{-\left(-39\right)±\sqrt{1521-560}}{2\times 2}$$

Add 1521 to -560.

$$x=\frac{-\left(-39\right)±\sqrt{961}}{2\times 2}$$

Take the square root of 961.

$$x=\frac{-\left(-39\right)±31}{2\times 2}$$

The opposite of -39 is 39.

$$x=\frac{39±31}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{39±31}{4}$$

Now solve the equation x=\frac{39±31}{4} when ± is plus. Add 39 to 31.

$$x=\frac{70}{4}$$

Reduce the fraction \frac{70}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{35}{2}$$

Now solve the equation x=\frac{39±31}{4} when ± is minus. Subtract 31 from 39.

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

Divide 8 by 4.

$$x=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{35}{2} for x_{1} and 2 for x_{2}.

$$2x^{2}-39x+70=2\left(x-\frac{35}{2}\right)\left(x-2\right)$$

Subtract \frac{35}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}-39x+70=2\times \frac{2x-35}{2}\left(x-2\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2x^{2}-39x+70=\left(2x-35\right)\left(x-2\right)$$

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 2

$$x ^ 2 -\frac{39}{2}x +35 = 0$$

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

$$r + s = \frac{39}{2} rs = 35$$

Two numbers r and s sum up to \frac{39}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{39}{2} = \frac{39}{4}. 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>

$$r = \frac{39}{4} - u s = \frac{39}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 35

$$(\frac{39}{4} - u) (\frac{39}{4} + u) = 35$$

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

$$\frac{1521}{16} - u^2 = 35$$

Simplify the expression by subtracting \frac{1521}{16} on both sides

$$-u^2 = 35-\frac{1521}{16} = -\frac{961}{16}$$

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

$$u^2 = \frac{961}{16} u = \pm\sqrt{\frac{961}{16}} = \pm \frac{31}{4}$$

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

$$r =\frac{39}{4} - \frac{31}{4} = 2 s = \frac{39}{4} + \frac{31}{4} = 17.500$$

Evaluate

$$\left(x-2\right)\left(2x-35\right)$$