Solve 9x^2+19x+10

$$9 x ^ { 2 } + 19 x + 10$$

Factor

$$\left(x+1\right)\left(9x+10\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=19 ab=9\times 10=90$$

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 90.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Calculate the sum for each pair.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

The solution is the pair that gives sum 19.

$$a=9 b=10$$

Rewrite 9x^{2}+19x+10 as \left(9x^{2}+9x\right)+\left(10x+10\right).

$$\left(9x^{2}+9x\right)+\left(10x+10\right)$$

Factor out 9x in the first and 10 in the second group.

$$9x\left(x+1\right)+10\left(x+1\right)$$

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

$$\left(x+1\right)\left(9x+10\right)$$

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.

$$9x^{2}+19x+10=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 9\times 10}}{2\times 9}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 9\times 10}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-19±\sqrt{361-36\times 10}}{2\times 9}$$

Multiply -36 times 10.

$$x=\frac{-19±\sqrt{361-360}}{2\times 9}$$

Add 361 to -360.

$$x=\frac{-19±\sqrt{1}}{2\times 9}$$

Take the square root of 1.

$$x=\frac{-19±1}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-19±1}{18}$$

Now solve the equation x=\frac{-19±1}{18} when ± is plus. Add -19 to 1.

$$x=-\frac{18}{18}$$

Divide -18 by 18.

$$x=-1$$

Now solve the equation x=\frac{-19±1}{18} when ± is minus. Subtract 1 from -19.

$$x=-\frac{20}{18}$$

Reduce the fraction \frac{-20}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{10}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{10}{9} for x_{2}.

$$9x^{2}+19x+10=9\left(x-\left(-1\right)\right)\left(x-\left(-\frac{10}{9}\right)\right)$$

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

$$9x^{2}+19x+10=9\left(x+1\right)\left(x+\frac{10}{9}\right)$$

Add \frac{10}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}+19x+10=9\left(x+1\right)\times \frac{9x+10}{9}$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9x^{2}+19x+10=\left(x+1\right)\left(9x+10\right)$$

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 9

$$x ^ 2 +\frac{19}{9}x +\frac{10}{9} = 0$$

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

$$r + s = -\frac{19}{9} rs = \frac{10}{9}$$

Two numbers r and s sum up to -\frac{19}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{19}{9} = -\frac{19}{18}. 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>

$$r = -\frac{19}{18} - u s = -\frac{19}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{10}{9}

$$(-\frac{19}{18} - u) (-\frac{19}{18} + u) = \frac{10}{9}$$

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

$$\frac{361}{324} - u^2 = \frac{10}{9}$$

Simplify the expression by subtracting \frac{361}{324} on both sides

$$-u^2 = \frac{10}{9}-\frac{361}{324} = -\frac{1}{324}$$

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

$$u^2 = \frac{1}{324} u = \pm\sqrt{\frac{1}{324}} = \pm \frac{1}{18}$$

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

$$r =-\frac{19}{18} - \frac{1}{18} = -1.111 s = -\frac{19}{18} + \frac{1}{18} = -1.000$$

Evaluate

$$\left(x+1\right)\left(9x+10\right)$$