Solve 9x^2+x-148

$$9 x ^ { 2 } + x - 148$$

Factor

$$\left(x-4\right)\left(9x+37\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx-148. To find a and b, set up a system to be solved.

$$a+b=1 ab=9\left(-148\right)=-1332$$

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 -1332.

$$-1,1332 -2,666 -3,444 -4,333 -6,222 -9,148 -12,111 -18,74 -36,37$$

Calculate the sum for each pair.

$$-1+1332=1331 -2+666=664 -3+444=441 -4+333=329 -6+222=216 -9+148=139 -12+111=99 -18+74=56 -36+37=1$$

The solution is the pair that gives sum 1.

$$a=-36 b=37$$

Rewrite 9x^{2}+x-148 as \left(9x^{2}-36x\right)+\left(37x-148\right).

$$\left(9x^{2}-36x\right)+\left(37x-148\right)$$

Factor out 9x in the first and 37 in the second group.

$$9x\left(x-4\right)+37\left(x-4\right)$$

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

$$\left(x-4\right)\left(9x+37\right)$$

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.

$$9x^{2}+x-148=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 9\left(-148\right)}}{2\times 9}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 9\left(-148\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-1±\sqrt{1-36\left(-148\right)}}{2\times 9}$$

Multiply -36 times -148.

$$x=\frac{-1±\sqrt{1+5328}}{2\times 9}$$

Add 1 to 5328.

$$x=\frac{-1±\sqrt{5329}}{2\times 9}$$

Take the square root of 5329.

$$x=\frac{-1±73}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-1±73}{18}$$

Now solve the equation x=\frac{-1±73}{18} when ± is plus. Add -1 to 73.

$$x=\frac{72}{18}$$

Divide 72 by 18.

$$x=4$$

Now solve the equation x=\frac{-1±73}{18} when ± is minus. Subtract 73 from -1.

$$x=-\frac{74}{18}$$

Reduce the fraction \frac{-74}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{37}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and -\frac{37}{9} for x_{2}.

$$9x^{2}+x-148=9\left(x-4\right)\left(x-\left(-\frac{37}{9}\right)\right)$$

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

$$9x^{2}+x-148=9\left(x-4\right)\left(x+\frac{37}{9}\right)$$

Add \frac{37}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}+x-148=9\left(x-4\right)\times \frac{9x+37}{9}$$

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

$$9x^{2}+x-148=\left(x-4\right)\left(9x+37\right)$$

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

$$x ^ 2 +\frac{1}{9}x -\frac{148}{9} = 0$$

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

$$r + s = -\frac{1}{9} rs = -\frac{148}{9}$$

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

$$r = -\frac{1}{18} - u s = -\frac{1}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{148}{9}

$$(-\frac{1}{18} - u) (-\frac{1}{18} + u) = -\frac{148}{9}$$

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

$$\frac{1}{324} - u^2 = -\frac{148}{9}$$

Simplify the expression by subtracting \frac{1}{324} on both sides

$$-u^2 = -\frac{148}{9}-\frac{1}{324} = -\frac{5329}{324}$$

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

$$u^2 = \frac{5329}{324} u = \pm\sqrt{\frac{5329}{324}} = \pm \frac{73}{18}$$

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

$$r =-\frac{1}{18} - \frac{73}{18} = -4.111 s = -\frac{1}{18} + \frac{73}{18} = 4$$

Evaluate

$$\left(x-4\right)\left(9x+37\right)$$