Solve 16x^2+17x-30

$$16 x ^ { 2 } + 17 x - 30$$

Factor

$$\left(16x-15\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=17 ab=16\left(-30\right)=-480$$

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 -480.

$$-1,480 -2,240 -3,160 -4,120 -5,96 -6,80 -8,60 -10,48 -12,40 -15,32 -16,30 -20,24$$

Calculate the sum for each pair.

$$-1+480=479 -2+240=238 -3+160=157 -4+120=116 -5+96=91 -6+80=74 -8+60=52 -10+48=38 -12+40=28 -15+32=17 -16+30=14 -20+24=4$$

The solution is the pair that gives sum 17.

$$a=-15 b=32$$

Rewrite 16x^{2}+17x-30 as \left(16x^{2}-15x\right)+\left(32x-30\right).

$$\left(16x^{2}-15x\right)+\left(32x-30\right)$$

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

$$x\left(16x-15\right)+2\left(16x-15\right)$$

Factor out common term 16x-15 by using distributive property.

$$\left(16x-15\right)\left(x+2\right)$$

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.

$$16x^{2}+17x-30=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 16\left(-30\right)}}{2\times 16}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 16\left(-30\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-17±\sqrt{289-64\left(-30\right)}}{2\times 16}$$

Multiply -64 times -30.

$$x=\frac{-17±\sqrt{289+1920}}{2\times 16}$$

Add 289 to 1920.

$$x=\frac{-17±\sqrt{2209}}{2\times 16}$$

Take the square root of 2209.

$$x=\frac{-17±47}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-17±47}{32}$$

Now solve the equation x=\frac{-17±47}{32} when ± is plus. Add -17 to 47.

$$x=\frac{30}{32}$$

Reduce the fraction \frac{30}{32} to lowest terms by extracting and canceling out 2.

$$x=\frac{15}{16}$$

Now solve the equation x=\frac{-17±47}{32} when ± is minus. Subtract 47 from -17.

$$x=-\frac{64}{32}$$

Divide -64 by 32.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15}{16} for x_{1} and -2 for x_{2}.

$$16x^{2}+17x-30=16\left(x-\frac{15}{16}\right)\left(x-\left(-2\right)\right)$$

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

$$16x^{2}+17x-30=16\left(x-\frac{15}{16}\right)\left(x+2\right)$$

Subtract \frac{15}{16} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+17x-30=16\times \frac{16x-15}{16}\left(x+2\right)$$

Cancel out 16, the greatest common factor in 16 and 16.

$$16x^{2}+17x-30=\left(16x-15\right)\left(x+2\right)$$

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 16

$$x ^ 2 +\frac{17}{16}x -\frac{15}{8} = 0$$

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

$$r + s = -\frac{17}{16} rs = -\frac{15}{8}$$

Two numbers r and s sum up to -\frac{17}{16} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{16} = -\frac{17}{32}. 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>

$$r = -\frac{17}{32} - u s = -\frac{17}{32} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{15}{8}

$$(-\frac{17}{32} - u) (-\frac{17}{32} + u) = -\frac{15}{8}$$

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

$$\frac{289}{1024} - u^2 = -\frac{15}{8}$$

Simplify the expression by subtracting \frac{289}{1024} on both sides

$$-u^2 = -\frac{15}{8}-\frac{289}{1024} = -\frac{2209}{1024}$$

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

$$u^2 = \frac{2209}{1024} u = \pm\sqrt{\frac{2209}{1024}} = \pm \frac{47}{32}$$

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

$$r =-\frac{17}{32} - \frac{47}{32} = -2 s = -\frac{17}{32} + \frac{47}{32} = 0.938$$

Evaluate

$$\left(16x-15\right)\left(x+2\right)$$