Solve -8x^2-12x+20rightarrow

$$- 8 x ^ { 2 } - 12 x + 20 \rightarrow$$

Factor

$$4\left(1-x\right)\left(2x+5\right)$$

Solution Steps

Factor out 4.

$$4\left(-2x^{2}-3x+5\right)$$

Consider -2x^{2}-3x+5. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-2\times 5=-10$$

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 -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -3.

$$a=2 b=-5$$

Rewrite -2x^{2}-3x+5 as \left(-2x^{2}+2x\right)+\left(-5x+5\right).

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

Factor out 2x in the first and 5 in the second group.

$$2x\left(-x+1\right)+5\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(2x+5\right)$$

Rewrite the complete factored expression.

$$4\left(-x+1\right)\left(2x+5\right)$$

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.

$$-8x^{2}-12x+20=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-8\right)\times 20}}{2\left(-8\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-8\right)\times 20}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-12\right)±\sqrt{144+32\times 20}}{2\left(-8\right)}$$

Multiply 32 times 20.

$$x=\frac{-\left(-12\right)±\sqrt{144+640}}{2\left(-8\right)}$$

Add 144 to 640.

$$x=\frac{-\left(-12\right)±\sqrt{784}}{2\left(-8\right)}$$

Take the square root of 784.

$$x=\frac{-\left(-12\right)±28}{2\left(-8\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±28}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{12±28}{-16}$$

Now solve the equation x=\frac{12±28}{-16} when ± is plus. Add 12 to 28.

$$x=\frac{40}{-16}$$

Reduce the fraction \frac{40}{-16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{5}{2}$$

Now solve the equation x=\frac{12±28}{-16} when ± is minus. Subtract 28 from 12.

$$x=-\frac{16}{-16}$$

Divide -16 by -16.

$$x=1$$

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

$$-8x^{2}-12x+20=-8\left(x-\left(-\frac{5}{2}\right)\right)\left(x-1\right)$$

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

$$-8x^{2}-12x+20=-8\left(x+\frac{5}{2}\right)\left(x-1\right)$$

Add \frac{5}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-8x^{2}-12x+20=-8\times \frac{-2x-5}{-2}\left(x-1\right)$$

Cancel out 2, the greatest common factor in -8 and 2.

$$-8x^{2}-12x+20=4\left(-2x-5\right)\left(x-1\right)$$

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.

$$x ^ 2 +\frac{3}{2}x -\frac{5}{2} = 0$$

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

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

Two numbers r and s sum up to -\frac{3}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{2} = -\frac{3}{4}. 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>

$$r = -\frac{3}{4} - u s = -\frac{3}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{2}

$$(-\frac{3}{4} - u) (-\frac{3}{4} + u) = -\frac{5}{2}$$

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

$$\frac{9}{16} - u^2 = -\frac{5}{2}$$

Simplify the expression by subtracting \frac{9}{16} on both sides

$$-u^2 = -\frac{5}{2}-\frac{9}{16} = -\frac{49}{16}$$

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

$$u^2 = \frac{49}{16} u = \pm\sqrt{\frac{49}{16}} = \pm \frac{7}{4}$$

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

$$r =-\frac{3}{4} - \frac{7}{4} = -2.500 s = -\frac{3}{4} + \frac{7}{4} = 1$$

Evaluate

$$20-12x-8x^{2}$$