Solve 72x^2+5x-25

$$72 x ^ { 2 } + 5 x - 25$$

Factor

$$\left(9x-5\right)\left(8x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 72x^{2}+ax+bx-25. To find a and b, set up a system to be solved.

$$a+b=5 ab=72\left(-25\right)=-1800$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1800.

$$-1,1800 -2,900 -3,600 -4,450 -5,360 -6,300 -8,225 -9,200 -10,180 -12,150 -15,120 -18,100 -20,90 -24,75 -25,72 -30,60 -36,50 -40,45$$

Calculate the sum for each pair.

$$-1+1800=1799 -2+900=898 -3+600=597 -4+450=446 -5+360=355 -6+300=294 -8+225=217 -9+200=191 -10+180=170 -12+150=138 -15+120=105 -18+100=82 -20+90=70 -24+75=51 -25+72=47 -30+60=30 -36+50=14 -40+45=5$$

The solution is the pair that gives sum 5.

$$a=-40 b=45$$

Rewrite 72x^{2}+5x-25 as \left(72x^{2}-40x\right)+\left(45x-25\right).

$$\left(72x^{2}-40x\right)+\left(45x-25\right)$$

Factor out 8x in the first and 5 in the second group.

$$8x\left(9x-5\right)+5\left(9x-5\right)$$

Factor out common term 9x-5 by using distributive property.

$$\left(9x-5\right)\left(8x+5\right)$$

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.

$$72x^{2}+5x-25=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 72\left(-25\right)}}{2\times 72}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 72\left(-25\right)}}{2\times 72}$$

Multiply -4 times 72.

$$x=\frac{-5±\sqrt{25-288\left(-25\right)}}{2\times 72}$$

Multiply -288 times -25.

$$x=\frac{-5±\sqrt{25+7200}}{2\times 72}$$

Add 25 to 7200.

$$x=\frac{-5±\sqrt{7225}}{2\times 72}$$

Take the square root of 7225.

$$x=\frac{-5±85}{2\times 72}$$

Multiply 2 times 72.

$$x=\frac{-5±85}{144}$$

Now solve the equation x=\frac{-5±85}{144} when ± is plus. Add -5 to 85.

$$x=\frac{80}{144}$$

Reduce the fraction \frac{80}{144} to lowest terms by extracting and canceling out 16.

$$x=\frac{5}{9}$$

Now solve the equation x=\frac{-5±85}{144} when ± is minus. Subtract 85 from -5.

$$x=-\frac{90}{144}$$

Reduce the fraction \frac{-90}{144} to lowest terms by extracting and canceling out 18.

$$x=-\frac{5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{9} for x_{1} and -\frac{5}{8} for x_{2}.

$$72x^{2}+5x-25=72\left(x-\frac{5}{9}\right)\left(x-\left(-\frac{5}{8}\right)\right)$$

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

$$72x^{2}+5x-25=72\left(x-\frac{5}{9}\right)\left(x+\frac{5}{8}\right)$$

Subtract \frac{5}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$72x^{2}+5x-25=72\times \frac{9x-5}{9}\left(x+\frac{5}{8}\right)$$

Add \frac{5}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$72x^{2}+5x-25=72\times \frac{9x-5}{9}\times \frac{8x+5}{8}$$

Multiply \frac{9x-5}{9} times \frac{8x+5}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$72x^{2}+5x-25=72\times \frac{\left(9x-5\right)\left(8x+5\right)}{9\times 8}$$

Multiply 9 times 8.

$$72x^{2}+5x-25=72\times \frac{\left(9x-5\right)\left(8x+5\right)}{72}$$

Cancel out 72, the greatest common factor in 72 and 72.

$$72x^{2}+5x-25=\left(9x-5\right)\left(8x+5\right)$$

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 72

$$x ^ 2 +\frac{5}{72}x -\frac{25}{72} = 0$$

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

$$r + s = -\frac{5}{72} rs = -\frac{25}{72}$$

Two numbers r and s sum up to -\frac{5}{72} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{72} = -\frac{5}{144}. 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>

$$r = -\frac{5}{144} - u s = -\frac{5}{144} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{25}{72}

$$(-\frac{5}{144} - u) (-\frac{5}{144} + u) = -\frac{25}{72}$$

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

$$\frac{25}{20736} - u^2 = -\frac{25}{72}$$

Simplify the expression by subtracting \frac{25}{20736} on both sides

$$-u^2 = -\frac{25}{72}-\frac{25}{20736} = -\frac{7225}{20736}$$

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

$$u^2 = \frac{7225}{20736} u = \pm\sqrt{\frac{7225}{20736}} = \pm \frac{85}{144}$$

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

$$r =-\frac{5}{144} - \frac{85}{144} = -0.625 s = -\frac{5}{144} + \frac{85}{144} = 0.556$$

Evaluate

$$\left(9x-5\right)\left(8x+5\right)$$