Solve 9x^2+17x-2

$$9 x ^ { 2 } + 17 x - 2$$

Factor

$$\left(9x-1\right)\left(x+2\right)$$

Solution Steps

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

$$a+b=17 ab=9\left(-2\right)=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 17.

$$a=-1 b=18$$

Rewrite 9x^{2}+17x-2 as \left(9x^{2}-x\right)+\left(18x-2\right).

$$\left(9x^{2}-x\right)+\left(18x-2\right)$$

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

$$x\left(9x-1\right)+2\left(9x-1\right)$$

Factor out common term 9x-1 by using distributive property.

$$\left(9x-1\right)\left(x+2\right)$$

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.

$$9x^{2}+17x-2=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 9\left(-2\right)}}{2\times 9}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 9\left(-2\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-17±\sqrt{289-36\left(-2\right)}}{2\times 9}$$

Multiply -36 times -2.

$$x=\frac{-17±\sqrt{289+72}}{2\times 9}$$

Add 289 to 72.

$$x=\frac{-17±\sqrt{361}}{2\times 9}$$

Take the square root of 361.

$$x=\frac{-17±19}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-17±19}{18}$$

Now solve the equation x=\frac{-17±19}{18} when ± is plus. Add -17 to 19.

$$x=\frac{2}{18}$$

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

$$x=\frac{1}{9}$$

Now solve the equation x=\frac{-17±19}{18} when ± is minus. Subtract 19 from -17.

$$x=-\frac{36}{18}$$

Divide -36 by 18.

$$x=-2$$

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

$$9x^{2}+17x-2=9\left(x-\frac{1}{9}\right)\left(x-\left(-2\right)\right)$$

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

$$9x^{2}+17x-2=9\left(x-\frac{1}{9}\right)\left(x+2\right)$$

Subtract \frac{1}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}+17x-2=9\times \frac{9x-1}{9}\left(x+2\right)$$

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

$$9x^{2}+17x-2=\left(9x-1\right)\left(x+2\right)$$

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

$$x ^ 2 +\frac{17}{9}x -\frac{2}{9} = 0$$

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

$$r + s = -\frac{17}{9} rs = -\frac{2}{9}$$

Two numbers r and s sum up to -\frac{17}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{9} = -\frac{17}{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>

$$r = -\frac{17}{18} - u s = -\frac{17}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{9}

$$(-\frac{17}{18} - u) (-\frac{17}{18} + u) = -\frac{2}{9}$$

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

$$\frac{289}{324} - u^2 = -\frac{2}{9}$$

Simplify the expression by subtracting \frac{289}{324} on both sides

$$-u^2 = -\frac{2}{9}-\frac{289}{324} = -\frac{361}{324}$$

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

$$u^2 = \frac{361}{324} u = \pm\sqrt{\frac{361}{324}} = \pm \frac{19}{18}$$

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

$$r =-\frac{17}{18} - \frac{19}{18} = -2 s = -\frac{17}{18} + \frac{19}{18} = 0.111$$

Evaluate

$$\left(9x-1\right)\left(x+2\right)$$