Solve -5m^2-6m+8=

$$- 5 m ^ { 2 } - 6 m + 8 =$$

Factor

$$-\left(5m-4\right)\left(m+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -5m^{2}+am+bm+8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-5\times 8=-40$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -6.

$$a=4 b=-10$$

Rewrite -5m^{2}-6m+8 as \left(-5m^{2}+4m\right)+\left(-10m+8\right).

$$\left(-5m^{2}+4m\right)+\left(-10m+8\right)$$

Factor out -m in the first and -2 in the second group.

$$-m\left(5m-4\right)-2\left(5m-4\right)$$

Factor out common term 5m-4 by using distributive property.

$$\left(5m-4\right)\left(-m-2\right)$$

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.

$$-5m^{2}-6m+8=0$$

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.

$$m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-5\right)\times 8}}{2\left(-5\right)}$$

Square -6.

$$m=\frac{-\left(-6\right)±\sqrt{36-4\left(-5\right)\times 8}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$m=\frac{-\left(-6\right)±\sqrt{36+20\times 8}}{2\left(-5\right)}$$

Multiply 20 times 8.

$$m=\frac{-\left(-6\right)±\sqrt{36+160}}{2\left(-5\right)}$$

Add 36 to 160.

$$m=\frac{-\left(-6\right)±\sqrt{196}}{2\left(-5\right)}$$

Take the square root of 196.

$$m=\frac{-\left(-6\right)±14}{2\left(-5\right)}$$

The opposite of -6 is 6.

$$m=\frac{6±14}{2\left(-5\right)}$$

Multiply 2 times -5.

$$m=\frac{6±14}{-10}$$

Now solve the equation m=\frac{6±14}{-10} when ± is plus. Add 6 to 14.

$$m=\frac{20}{-10}$$

Divide 20 by -10.

$$m=-2$$

Now solve the equation m=\frac{6±14}{-10} when ± is minus. Subtract 14 from 6.

$$m=-\frac{8}{-10}$$

Reduce the fraction \frac{-8}{-10} to lowest terms by extracting and canceling out 2.

$$m=\frac{4}{5}$$

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

$$-5m^{2}-6m+8=-5\left(m-\left(-2\right)\right)\left(m-\frac{4}{5}\right)$$

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

$$-5m^{2}-6m+8=-5\left(m+2\right)\left(m-\frac{4}{5}\right)$$

Subtract \frac{4}{5} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-5m^{2}-6m+8=-5\left(m+2\right)\times \frac{-5m+4}{-5}$$

Cancel out 5, the greatest common factor in -5 and 5.

$$-5m^{2}-6m+8=\left(m+2\right)\left(-5m+4\right)$$

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.

$$x ^ 2 +\frac{6}{5}x -\frac{8}{5} = 0$$

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

$$r + s = -\frac{6}{5} rs = -\frac{8}{5}$$

Two numbers r and s sum up to -\frac{6}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{6}{5} = -\frac{3}{5}. 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>

$$r = -\frac{3}{5} - u s = -\frac{3}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{5}

$$(-\frac{3}{5} - u) (-\frac{3}{5} + u) = -\frac{8}{5}$$

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

$$\frac{9}{25} - u^2 = -\frac{8}{5}$$

Simplify the expression by subtracting \frac{9}{25} on both sides

$$-u^2 = -\frac{8}{5}-\frac{9}{25} = -\frac{49}{25}$$

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

$$u^2 = \frac{49}{25} u = \pm\sqrt{\frac{49}{25}} = \pm \frac{7}{5}$$

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

$$r =-\frac{3}{5} - \frac{7}{5} = -2 s = -\frac{3}{5} + \frac{7}{5} = 0.800$$

Evaluate

$$-\left(5m-4\right)\left(m+2\right)$$