Solve (22+51x-10x^2)_{max}

$$( 22 + 51 x - 10 x ^ { 2 } ) _ { \max }$$

Factor

$$-\left(2x-11\right)\left(5x+2\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-10x^{2}+51x+22$$

Factor the expression by grouping. First, the expression needs to be rewritten as -10x^{2}+ax+bx+22. To find a and b, set up a system to be solved.

$$a+b=51 ab=-10\times 22=-220$$

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 -220.

$$-1,220 -2,110 -4,55 -5,44 -10,22 -11,20$$

Calculate the sum for each pair.

$$-1+220=219 -2+110=108 -4+55=51 -5+44=39 -10+22=12 -11+20=9$$

The solution is the pair that gives sum 51.

$$a=55 b=-4$$

Rewrite -10x^{2}+51x+22 as \left(-10x^{2}+55x\right)+\left(-4x+22\right).

$$\left(-10x^{2}+55x\right)+\left(-4x+22\right)$$

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

$$-5x\left(2x-11\right)-2\left(2x-11\right)$$

Factor out common term 2x-11 by using distributive property.

$$\left(2x-11\right)\left(-5x-2\right)$$

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.

$$-10x^{2}+51x+22=0$$

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.

$$x=\frac{-51±\sqrt{51^{2}-4\left(-10\right)\times 22}}{2\left(-10\right)}$$

Square 51.

$$x=\frac{-51±\sqrt{2601-4\left(-10\right)\times 22}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-51±\sqrt{2601+40\times 22}}{2\left(-10\right)}$$

Multiply 40 times 22.

$$x=\frac{-51±\sqrt{2601+880}}{2\left(-10\right)}$$

Add 2601 to 880.

$$x=\frac{-51±\sqrt{3481}}{2\left(-10\right)}$$

Take the square root of 3481.

$$x=\frac{-51±59}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-51±59}{-20}$$

Now solve the equation x=\frac{-51±59}{-20} when ± is plus. Add -51 to 59.

$$x=\frac{8}{-20}$$

Reduce the fraction \frac{8}{-20} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-51±59}{-20} when ± is minus. Subtract 59 from -51.

$$x=-\frac{110}{-20}$$

Reduce the fraction \frac{-110}{-20} to lowest terms by extracting and canceling out 10.

$$x=\frac{11}{2}$$

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 \frac{11}{2} for x_{2}.

$$-10x^{2}+51x+22=-10\left(x-\left(-\frac{2}{5}\right)\right)\left(x-\frac{11}{2}\right)$$

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

$$-10x^{2}+51x+22=-10\left(x+\frac{2}{5}\right)\left(x-\frac{11}{2}\right)$$

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

$$-10x^{2}+51x+22=-10\times \frac{-5x-2}{-5}\left(x-\frac{11}{2}\right)$$

Subtract \frac{11}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-10x^{2}+51x+22=-10\times \frac{-5x-2}{-5}\times \frac{-2x+11}{-2}$$

Multiply \frac{-5x-2}{-5} times \frac{-2x+11}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-10x^{2}+51x+22=-10\times \frac{\left(-5x-2\right)\left(-2x+11\right)}{-5\left(-2\right)}$$

Multiply -5 times -2.

$$-10x^{2}+51x+22=-10\times \frac{\left(-5x-2\right)\left(-2x+11\right)}{10}$$

Cancel out 10, the greatest common factor in -10 and 10.

$$-10x^{2}+51x+22=-\left(-5x-2\right)\left(-2x+11\right)$$

Evaluate

$$22+51x-10x^{2}$$