Solve -27m^2-51m-24

$$- 27 m ^ { 2 } - 51 m - 24$$

Factor

$$3\left(-m-1\right)\left(9m+8\right)$$

Solution Steps

Factor out 3.

$$3\left(-9m^{2}-17m-8\right)$$

Consider -9m^{2}-17m-8. Factor the expression by grouping. First, the expression needs to be rewritten as -9m^{2}+am+bm-8. To find a and b, set up a system to be solved.

$$a+b=-17 ab=-9\left(-8\right)=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -17.

$$a=-8 b=-9$$

Rewrite -9m^{2}-17m-8 as \left(-9m^{2}-8m\right)+\left(-9m-8\right).

$$\left(-9m^{2}-8m\right)+\left(-9m-8\right)$$

Factor out -m in the first and -1 in the second group.

$$-m\left(9m+8\right)-\left(9m+8\right)$$

Factor out common term 9m+8 by using distributive property.

$$\left(9m+8\right)\left(-m-1\right)$$

Rewrite the complete factored expression.

$$3\left(9m+8\right)\left(-m-1\right)$$

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.

$$-27m^{2}-51m-24=0$$

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.

$$m=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\left(-27\right)\left(-24\right)}}{2\left(-27\right)}$$

Square -51.

$$m=\frac{-\left(-51\right)±\sqrt{2601-4\left(-27\right)\left(-24\right)}}{2\left(-27\right)}$$

Multiply -4 times -27.

$$m=\frac{-\left(-51\right)±\sqrt{2601+108\left(-24\right)}}{2\left(-27\right)}$$

Multiply 108 times -24.

$$m=\frac{-\left(-51\right)±\sqrt{2601-2592}}{2\left(-27\right)}$$

Add 2601 to -2592.

$$m=\frac{-\left(-51\right)±\sqrt{9}}{2\left(-27\right)}$$

Take the square root of 9.

$$m=\frac{-\left(-51\right)±3}{2\left(-27\right)}$$

The opposite of -51 is 51.

$$m=\frac{51±3}{2\left(-27\right)}$$

Multiply 2 times -27.

$$m=\frac{51±3}{-54}$$

Now solve the equation m=\frac{51±3}{-54} when ± is plus. Add 51 to 3.

$$m=\frac{54}{-54}$$

Divide 54 by -54.

$$m=-1$$

Now solve the equation m=\frac{51±3}{-54} when ± is minus. Subtract 3 from 51.

$$m=\frac{48}{-54}$$

Reduce the fraction \frac{48}{-54} to lowest terms by extracting and canceling out 6.

$$m=-\frac{8}{9}$$

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

$$-27m^{2}-51m-24=-27\left(m-\left(-1\right)\right)\left(m-\left(-\frac{8}{9}\right)\right)$$

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

$$-27m^{2}-51m-24=-27\left(m+1\right)\left(m+\frac{8}{9}\right)$$

Add \frac{8}{9} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-27m^{2}-51m-24=-27\left(m+1\right)\times \frac{-9m-8}{-9}$$

Cancel out 9, the greatest common factor in -27 and 9.

$$-27m^{2}-51m-24=3\left(m+1\right)\left(-9m-8\right)$$

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.

$$x ^ 2 +\frac{17}{9}x +\frac{8}{9} = 0$$

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

$$r + s = -\frac{17}{9} rs = \frac{8}{9}$$

Two numbers r and s sum up to -\frac{17}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{9} = -\frac{17}{18}. 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>

$$r = -\frac{17}{18} - u s = -\frac{17}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{9}

$$(-\frac{17}{18} - u) (-\frac{17}{18} + u) = \frac{8}{9}$$

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

$$\frac{289}{324} - u^2 = \frac{8}{9}$$

Simplify the expression by subtracting \frac{289}{324} on both sides

$$-u^2 = \frac{8}{9}-\frac{289}{324} = -\frac{1}{324}$$

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

$$u^2 = \frac{1}{324} u = \pm\sqrt{\frac{1}{324}} = \pm \frac{1}{18}$$

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

$$r =-\frac{17}{18} - \frac{1}{18} = -1 s = -\frac{17}{18} + \frac{1}{18} = -0.889$$

Evaluate

$$-27m^{2}-51m-24$$