Solve -12x^2+59x+5

$$- 12 x ^ { 2 } + 59 x + 5$$

Factor

$$\left(5-x\right)\left(12x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -12x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=59 ab=-12\times 5=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 59.

$$a=60 b=-1$$

Rewrite -12x^{2}+59x+5 as \left(-12x^{2}+60x\right)+\left(-x+5\right).

$$\left(-12x^{2}+60x\right)+\left(-x+5\right)$$

Factor out 12x in -12x^{2}+60x.

$$12x\left(-x+5\right)-x+5$$

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

$$\left(-x+5\right)\left(12x+1\right)$$

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.

$$-12x^{2}+59x+5=0$$

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.

$$x=\frac{-59±\sqrt{59^{2}-4\left(-12\right)\times 5}}{2\left(-12\right)}$$

Square 59.

$$x=\frac{-59±\sqrt{3481-4\left(-12\right)\times 5}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-59±\sqrt{3481+48\times 5}}{2\left(-12\right)}$$

Multiply 48 times 5.

$$x=\frac{-59±\sqrt{3481+240}}{2\left(-12\right)}$$

Add 3481 to 240.

$$x=\frac{-59±\sqrt{3721}}{2\left(-12\right)}$$

Take the square root of 3721.

$$x=\frac{-59±61}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-59±61}{-24}$$

Now solve the equation x=\frac{-59±61}{-24} when ± is plus. Add -59 to 61.

$$x=\frac{2}{-24}$$

Reduce the fraction \frac{2}{-24} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{12}$$

Now solve the equation x=\frac{-59±61}{-24} when ± is minus. Subtract 61 from -59.

$$x=-\frac{120}{-24}$$

Divide -120 by -24.

$$x=5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{12} for x_{1} and 5 for x_{2}.

$$-12x^{2}+59x+5=-12\left(x-\left(-\frac{1}{12}\right)\right)\left(x-5\right)$$

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

$$-12x^{2}+59x+5=-12\left(x+\frac{1}{12}\right)\left(x-5\right)$$

Add \frac{1}{12} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-12x^{2}+59x+5=-12\times \frac{-12x-1}{-12}\left(x-5\right)$$

Cancel out 12, the greatest common factor in -12 and 12.

$$-12x^{2}+59x+5=\left(-12x-1\right)\left(x-5\right)$$

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.

$$x ^ 2 -\frac{59}{12}x -\frac{5}{12} = 0$$

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

$$r + s = \frac{59}{12} rs = -\frac{5}{12}$$

Two numbers r and s sum up to \frac{59}{12} exactly when the average of the two numbers is \frac{1}{2}*\frac{59}{12} = \frac{59}{24}. 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>

$$r = \frac{59}{24} - u s = \frac{59}{24} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{12}

$$(\frac{59}{24} - u) (\frac{59}{24} + u) = -\frac{5}{12}$$

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

$$\frac{3481}{576} - u^2 = -\frac{5}{12}$$

Simplify the expression by subtracting \frac{3481}{576} on both sides

$$-u^2 = -\frac{5}{12}-\frac{3481}{576} = -\frac{3721}{576}$$

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

$$u^2 = \frac{3721}{576} u = \pm\sqrt{\frac{3721}{576}} = \pm \frac{61}{24}$$

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

$$r =\frac{59}{24} - \frac{61}{24} = -0.083 s = \frac{59}{24} + \frac{61}{24} = 5$$

Evaluate

$$\left(5-x\right)\left(12x+1\right)$$