Solve 10x^2+3x-18

$$10 x ^ { 2 } + 3 x - 18$$

Factor

$$\left(5x-6\right)\left(2x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 10x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=3 ab=10\left(-18\right)=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Calculate the sum for each pair.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

The solution is the pair that gives sum 3.

$$a=-12 b=15$$

Rewrite 10x^{2}+3x-18 as \left(10x^{2}-12x\right)+\left(15x-18\right).

$$\left(10x^{2}-12x\right)+\left(15x-18\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(5x-6\right)+3\left(5x-6\right)$$

Factor out common term 5x-6 by using distributive property.

$$\left(5x-6\right)\left(2x+3\right)$$

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.

$$10x^{2}+3x-18=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 10\left(-18\right)}}{2\times 10}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 10\left(-18\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-3±\sqrt{9-40\left(-18\right)}}{2\times 10}$$

Multiply -40 times -18.

$$x=\frac{-3±\sqrt{9+720}}{2\times 10}$$

Add 9 to 720.

$$x=\frac{-3±\sqrt{729}}{2\times 10}$$

Take the square root of 729.

$$x=\frac{-3±27}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-3±27}{20}$$

Now solve the equation x=\frac{-3±27}{20} when ± is plus. Add -3 to 27.

$$x=\frac{24}{20}$$

Reduce the fraction \frac{24}{20} to lowest terms by extracting and canceling out 4.

$$x=\frac{6}{5}$$

Now solve the equation x=\frac{-3±27}{20} when ± is minus. Subtract 27 from -3.

$$x=-\frac{30}{20}$$

Reduce the fraction \frac{-30}{20} to lowest terms by extracting and canceling out 10.

$$x=-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{6}{5} for x_{1} and -\frac{3}{2} for x_{2}.

$$10x^{2}+3x-18=10\left(x-\frac{6}{5}\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$10x^{2}+3x-18=10\left(x-\frac{6}{5}\right)\left(x+\frac{3}{2}\right)$$

Subtract \frac{6}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+3x-18=10\times \frac{5x-6}{5}\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+3x-18=10\times \frac{5x-6}{5}\times \frac{2x+3}{2}$$

Multiply \frac{5x-6}{5} times \frac{2x+3}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$10x^{2}+3x-18=10\times \frac{\left(5x-6\right)\left(2x+3\right)}{5\times 2}$$

Multiply 5 times 2.

$$10x^{2}+3x-18=10\times \frac{\left(5x-6\right)\left(2x+3\right)}{10}$$

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

$$10x^{2}+3x-18=\left(5x-6\right)\left(2x+3\right)$$

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 10

$$x ^ 2 +\frac{3}{10}x -\frac{9}{5} = 0$$

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

$$r + s = -\frac{3}{10} rs = -\frac{9}{5}$$

Two numbers r and s sum up to -\frac{3}{10} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{10} = -\frac{3}{20}. 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>

$$r = -\frac{3}{20} - u s = -\frac{3}{20} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{5}

$$(-\frac{3}{20} - u) (-\frac{3}{20} + u) = -\frac{9}{5}$$

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

$$\frac{9}{400} - u^2 = -\frac{9}{5}$$

Simplify the expression by subtracting \frac{9}{400} on both sides

$$-u^2 = -\frac{9}{5}-\frac{9}{400} = -\frac{729}{400}$$

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

$$u^2 = \frac{729}{400} u = \pm\sqrt{\frac{729}{400}} = \pm \frac{27}{20}$$

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

$$r =-\frac{3}{20} - \frac{27}{20} = -1.500 s = -\frac{3}{20} + \frac{27}{20} = 1.200$$

Evaluate

$$\left(5x-6\right)\left(2x+3\right)$$