Solve 8x^2-230x+750

$$8 x ^ { 2 } - 230 x + 750$$

Factor

$$2\left(x-25\right)\left(4x-15\right)$$

Solution Steps

Factor out 2.

$$2\left(4x^{2}-115x+375\right)$$

Consider 4x^{2}-115x+375. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+375. To find a and b, set up a system to be solved.

$$a+b=-115 ab=4\times 375=1500$$

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 1500.

$$-1,-1500 -2,-750 -3,-500 -4,-375 -5,-300 -6,-250 -10,-150 -12,-125 -15,-100 -20,-75 -25,-60 -30,-50$$

Calculate the sum for each pair.

$$-1-1500=-1501 -2-750=-752 -3-500=-503 -4-375=-379 -5-300=-305 -6-250=-256 -10-150=-160 -12-125=-137 -15-100=-115 -20-75=-95 -25-60=-85 -30-50=-80$$

The solution is the pair that gives sum -115.

$$a=-100 b=-15$$

Rewrite 4x^{2}-115x+375 as \left(4x^{2}-100x\right)+\left(-15x+375\right).

$$\left(4x^{2}-100x\right)+\left(-15x+375\right)$$

Factor out 4x in the first and -15 in the second group.

$$4x\left(x-25\right)-15\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(4x-15\right)$$

Rewrite the complete factored expression.

$$2\left(x-25\right)\left(4x-15\right)$$

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.

$$8x^{2}-230x+750=0$$

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.

$$x=\frac{-\left(-230\right)±\sqrt{\left(-230\right)^{2}-4\times 8\times 750}}{2\times 8}$$

Square -230.

$$x=\frac{-\left(-230\right)±\sqrt{52900-4\times 8\times 750}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-230\right)±\sqrt{52900-32\times 750}}{2\times 8}$$

Multiply -32 times 750.

$$x=\frac{-\left(-230\right)±\sqrt{52900-24000}}{2\times 8}$$

Add 52900 to -24000.

$$x=\frac{-\left(-230\right)±\sqrt{28900}}{2\times 8}$$

Take the square root of 28900.

$$x=\frac{-\left(-230\right)±170}{2\times 8}$$

The opposite of -230 is 230.

$$x=\frac{230±170}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{230±170}{16}$$

Now solve the equation x=\frac{230±170}{16} when ± is plus. Add 230 to 170.

$$x=\frac{400}{16}$$

Divide 400 by 16.

$$x=25$$

Now solve the equation x=\frac{230±170}{16} when ± is minus. Subtract 170 from 230.

$$x=\frac{60}{16}$$

Reduce the fraction \frac{60}{16} to lowest terms by extracting and canceling out 4.

$$x=\frac{15}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 25 for x_{1} and \frac{15}{4} for x_{2}.

$$8x^{2}-230x+750=8\left(x-25\right)\left(x-\frac{15}{4}\right)$$

Subtract \frac{15}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}-230x+750=8\left(x-25\right)\times \frac{4x-15}{4}$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8x^{2}-230x+750=2\left(x-25\right)\left(4x-15\right)$$

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 8

$$x ^ 2 -\frac{115}{4}x +\frac{375}{4} = 0$$

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

$$r + s = \frac{115}{4} rs = \frac{375}{4}$$

Two numbers r and s sum up to \frac{115}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{115}{4} = \frac{115}{8}. 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>

$$r = \frac{115}{8} - u s = \frac{115}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{375}{4}

$$(\frac{115}{8} - u) (\frac{115}{8} + u) = \frac{375}{4}$$

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

$$\frac{13225}{64} - u^2 = \frac{375}{4}$$

Simplify the expression by subtracting \frac{13225}{64} on both sides

$$-u^2 = \frac{375}{4}-\frac{13225}{64} = -\frac{7225}{64}$$

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

$$u^2 = \frac{7225}{64} u = \pm\sqrt{\frac{7225}{64}} = \pm \frac{85}{8}$$

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

$$r =\frac{115}{8} - \frac{85}{8} = 3.750 s = \frac{115}{8} + \frac{85}{8} = 25$$

Evaluate

$$2\left(x-25\right)\left(4x-15\right)$$