Solve 22p^2+51p-10

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

Factor

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

Solution Steps

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

$$a+b=51 ab=22\left(-10\right)=-220$$

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.

$$a=-4 b=55$$

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

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

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

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

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

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

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.

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

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.

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

Square 51.

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

Multiply -4 times 22.

$$p=\frac{-51±\sqrt{2601-88\left(-10\right)}}{2\times 22}$$

Multiply -88 times -10.

$$p=\frac{-51±\sqrt{2601+880}}{2\times 22}$$

Add 2601 to 880.

$$p=\frac{-51±\sqrt{3481}}{2\times 22}$$

Take the square root of 3481.

$$p=\frac{-51±59}{2\times 22}$$

Multiply 2 times 22.

$$p=\frac{-51±59}{44}$$

Now solve the equation p=\frac{-51±59}{44} when ± is plus. Add -51 to 59.

$$p=\frac{8}{44}$$

Reduce the fraction \frac{8}{44} to lowest terms by extracting and canceling out 4.

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

Now solve the equation p=\frac{-51±59}{44} when ± is minus. Subtract 59 from -51.

$$p=-\frac{110}{44}$$

Reduce the fraction \frac{-110}{44} to lowest terms by extracting and canceling out 22.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{11} for x_{1} and -\frac{5}{2} for x_{2}.

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

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

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

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

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

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

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

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

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

Multiply 11 times 2.

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

Cancel out 22, the greatest common factor in 22 and 22.

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

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.This is achieved by dividing both sides of the equation by 22

$$x ^ 2 +\frac{51}{22}x -\frac{5}{11} = 0$$

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

$$r + s = -\frac{51}{22} rs = -\frac{5}{11}$$

Two numbers r and s sum up to -\frac{51}{22} exactly when the average of the two numbers is \frac{1}{2}*-\frac{51}{22} = -\frac{51}{44}. 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>

$$r = -\frac{51}{44} - u s = -\frac{51}{44} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{11}

$$(-\frac{51}{44} - u) (-\frac{51}{44} + u) = -\frac{5}{11}$$

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

$$\frac{2601}{1936} - u^2 = -\frac{5}{11}$$

Simplify the expression by subtracting \frac{2601}{1936} on both sides

$$-u^2 = -\frac{5}{11}-\frac{2601}{1936} = -\frac{3481}{1936}$$

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

$$u^2 = \frac{3481}{1936} u = \pm\sqrt{\frac{3481}{1936}} = \pm \frac{59}{44}$$

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

$$r =-\frac{51}{44} - \frac{59}{44} = -2.500 s = -\frac{51}{44} + \frac{59}{44} = 0.182$$

Evaluate

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