Solve -5x^2+22x-8

$$- 5 x ^ { 2 } + 22 x - 8$$

Factor

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

Solution Steps

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

$$a+b=22 ab=-5\left(-8\right)=40$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 40.

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

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 22.

$$a=20 b=2$$

Rewrite -5x^{2}+22x-8 as \left(-5x^{2}+20x\right)+\left(2x-8\right).

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

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

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

Factor out common term -x+4 by using distributive property.

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

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.

$$-5x^{2}+22x-8=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-8\right)}}{2\left(-5\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-5\right)\left(-8\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-22±\sqrt{484+20\left(-8\right)}}{2\left(-5\right)}$$

Multiply 20 times -8.

$$x=\frac{-22±\sqrt{484-160}}{2\left(-5\right)}$$

Add 484 to -160.

$$x=\frac{-22±\sqrt{324}}{2\left(-5\right)}$$

Take the square root of 324.

$$x=\frac{-22±18}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-22±18}{-10}$$

Now solve the equation x=\frac{-22±18}{-10} when ± is plus. Add -22 to 18.

$$x=-\frac{4}{-10}$$

Reduce the fraction \frac{-4}{-10} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-22±18}{-10} when ± is minus. Subtract 18 from -22.

$$x=-\frac{40}{-10}$$

Divide -40 by -10.

$$x=4$$

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

$$-5x^{2}+22x-8=-5\left(x-\frac{2}{5}\right)\left(x-4\right)$$

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

$$-5x^{2}+22x-8=-5\times \frac{-5x+2}{-5}\left(x-4\right)$$

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

$$-5x^{2}+22x-8=\left(-5x+2\right)\left(x-4\right)$$

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.

$$x ^ 2 -\frac{22}{5}x +\frac{8}{5} = 0$$

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

$$r + s = \frac{22}{5} rs = \frac{8}{5}$$

Two numbers r and s sum up to \frac{22}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{22}{5} = \frac{11}{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>

$$r = \frac{11}{5} - u s = \frac{11}{5} + u$$

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

$$(\frac{11}{5} - u) (\frac{11}{5} + u) = \frac{8}{5}$$

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

$$\frac{121}{25} - u^2 = \frac{8}{5}$$

Simplify the expression by subtracting \frac{121}{25} on both sides

$$-u^2 = \frac{8}{5}-\frac{121}{25} = -\frac{81}{25}$$

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

$$u^2 = \frac{81}{25} u = \pm\sqrt{\frac{81}{25}} = \pm \frac{9}{5}$$

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

$$r =\frac{11}{5} - \frac{9}{5} = 0.400 s = \frac{11}{5} + \frac{9}{5} = 4$$

Evaluate

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