Solve -60x^2+51x+72

$$- 60 x ^ { 2 } + 51 x + 72$$

Factor

$$3\left(8-5x\right)\left(4x+3\right)$$

Solution Steps

Factor out 3.

$$3\left(-20x^{2}+17x+24\right)$$

Consider -20x^{2}+17x+24. Factor the expression by grouping. First, the expression needs to be rewritten as -20x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=17 ab=-20\times 24=-480$$

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 -480.

$$-1,480 -2,240 -3,160 -4,120 -5,96 -6,80 -8,60 -10,48 -12,40 -15,32 -16,30 -20,24$$

Calculate the sum for each pair.

$$-1+480=479 -2+240=238 -3+160=157 -4+120=116 -5+96=91 -6+80=74 -8+60=52 -10+48=38 -12+40=28 -15+32=17 -16+30=14 -20+24=4$$

The solution is the pair that gives sum 17.

$$a=32 b=-15$$

Rewrite -20x^{2}+17x+24 as \left(-20x^{2}+32x\right)+\left(-15x+24\right).

$$\left(-20x^{2}+32x\right)+\left(-15x+24\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(-5x+8\right)+3\left(-5x+8\right)$$

Factor out common term -5x+8 by using distributive property.

$$\left(-5x+8\right)\left(4x+3\right)$$

Rewrite the complete factored expression.

$$3\left(-5x+8\right)\left(4x+3\right)$$

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.

$$-60x^{2}+51x+72=0$$

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.

$$x=\frac{-51±\sqrt{51^{2}-4\left(-60\right)\times 72}}{2\left(-60\right)}$$

Square 51.

$$x=\frac{-51±\sqrt{2601-4\left(-60\right)\times 72}}{2\left(-60\right)}$$

Multiply -4 times -60.

$$x=\frac{-51±\sqrt{2601+240\times 72}}{2\left(-60\right)}$$

Multiply 240 times 72.

$$x=\frac{-51±\sqrt{2601+17280}}{2\left(-60\right)}$$

Add 2601 to 17280.

$$x=\frac{-51±\sqrt{19881}}{2\left(-60\right)}$$

Take the square root of 19881.

$$x=\frac{-51±141}{2\left(-60\right)}$$

Multiply 2 times -60.

$$x=\frac{-51±141}{-120}$$

Now solve the equation x=\frac{-51±141}{-120} when ± is plus. Add -51 to 141.

$$x=\frac{90}{-120}$$

Reduce the fraction \frac{90}{-120} to lowest terms by extracting and canceling out 30.

$$x=-\frac{3}{4}$$

Now solve the equation x=\frac{-51±141}{-120} when ± is minus. Subtract 141 from -51.

$$x=-\frac{192}{-120}$$

Reduce the fraction \frac{-192}{-120} to lowest terms by extracting and canceling out 24.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{4} for x_{1} and \frac{8}{5} for x_{2}.

$$-60x^{2}+51x+72=-60\left(x-\left(-\frac{3}{4}\right)\right)\left(x-\frac{8}{5}\right)$$

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

$$-60x^{2}+51x+72=-60\left(x+\frac{3}{4}\right)\left(x-\frac{8}{5}\right)$$

Add \frac{3}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-60x^{2}+51x+72=-60\times \frac{-4x-3}{-4}\left(x-\frac{8}{5}\right)$$

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

$$-60x^{2}+51x+72=-60\times \frac{-4x-3}{-4}\times \frac{-5x+8}{-5}$$

Multiply \frac{-4x-3}{-4} times \frac{-5x+8}{-5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-60x^{2}+51x+72=-60\times \frac{\left(-4x-3\right)\left(-5x+8\right)}{-4\left(-5\right)}$$

Multiply -4 times -5.

$$-60x^{2}+51x+72=-60\times \frac{\left(-4x-3\right)\left(-5x+8\right)}{20}$$

Cancel out 20, the greatest common factor in -60 and 20.

$$-60x^{2}+51x+72=-3\left(-4x-3\right)\left(-5x+8\right)$$

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.

$$x ^ 2 -\frac{17}{20}x -\frac{6}{5} = 0$$

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

$$r + s = \frac{17}{20} rs = -\frac{6}{5}$$

Two numbers r and s sum up to \frac{17}{20} exactly when the average of the two numbers is \frac{1}{2}*\frac{17}{20} = \frac{17}{40}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{17}{40} - u s = \frac{17}{40} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{6}{5}

$$(\frac{17}{40} - u) (\frac{17}{40} + u) = -\frac{6}{5}$$

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

$$\frac{289}{1600} - u^2 = -\frac{6}{5}$$

Simplify the expression by subtracting \frac{289}{1600} on both sides

$$-u^2 = -\frac{6}{5}-\frac{289}{1600} = -\frac{2209}{1600}$$

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

$$u^2 = \frac{2209}{1600} u = \pm\sqrt{\frac{2209}{1600}} = \pm \frac{47}{40}$$

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

$$r =\frac{17}{40} - \frac{47}{40} = -0.750 s = \frac{17}{40} + \frac{47}{40} = 1.600$$

Evaluate

$$72+51x-60x^{2}$$