Solve 16x^2-46x+15

$$16 x ^ { 2 } - 46 x + 15$$

Factor

$$\left(2x-5\right)\left(8x-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-46 ab=16\times 15=240$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 240.

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

Calculate the sum for each pair.

$$-1-240=-241 -2-120=-122 -3-80=-83 -4-60=-64 -5-48=-53 -6-40=-46 -8-30=-38 -10-24=-34 -12-20=-32 -15-16=-31$$

The solution is the pair that gives sum -46.

$$a=-40 b=-6$$

Rewrite 16x^{2}-46x+15 as \left(16x^{2}-40x\right)+\left(-6x+15\right).

$$\left(16x^{2}-40x\right)+\left(-6x+15\right)$$

Factor out 8x in the first and -3 in the second group.

$$8x\left(2x-5\right)-3\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(8x-3\right)$$

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.

$$16x^{2}-46x+15=0$$

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.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\times 16\times 15}}{2\times 16}$$

Square -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\times 16\times 15}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-46\right)±\sqrt{2116-64\times 15}}{2\times 16}$$

Multiply -64 times 15.

$$x=\frac{-\left(-46\right)±\sqrt{2116-960}}{2\times 16}$$

Add 2116 to -960.

$$x=\frac{-\left(-46\right)±\sqrt{1156}}{2\times 16}$$

Take the square root of 1156.

$$x=\frac{-\left(-46\right)±34}{2\times 16}$$

The opposite of -46 is 46.

$$x=\frac{46±34}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{46±34}{32}$$

Now solve the equation x=\frac{46±34}{32} when ± is plus. Add 46 to 34.

$$x=\frac{80}{32}$$

Reduce the fraction \frac{80}{32} to lowest terms by extracting and canceling out 16.

$$x=\frac{5}{2}$$

Now solve the equation x=\frac{46±34}{32} when ± is minus. Subtract 34 from 46.

$$x=\frac{12}{32}$$

Reduce the fraction \frac{12}{32} to lowest terms by extracting and canceling out 4.

$$x=\frac{3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{2} for x_{1} and \frac{3}{8} for x_{2}.

$$16x^{2}-46x+15=16\left(x-\frac{5}{2}\right)\left(x-\frac{3}{8}\right)$$

Subtract \frac{5}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}-46x+15=16\times \frac{2x-5}{2}\left(x-\frac{3}{8}\right)$$

Subtract \frac{3}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}-46x+15=16\times \frac{2x-5}{2}\times \frac{8x-3}{8}$$

Multiply \frac{2x-5}{2} times \frac{8x-3}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$16x^{2}-46x+15=16\times \frac{\left(2x-5\right)\left(8x-3\right)}{2\times 8}$$

Multiply 2 times 8.

$$16x^{2}-46x+15=16\times \frac{\left(2x-5\right)\left(8x-3\right)}{16}$$

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

$$16x^{2}-46x+15=\left(2x-5\right)\left(8x-3\right)$$

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 16

$$x ^ 2 -\frac{23}{8}x +\frac{15}{16} = 0$$

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

$$r + s = \frac{23}{8} rs = \frac{15}{16}$$

Two numbers r and s sum up to \frac{23}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{8} = \frac{23}{16}. 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>

$$r = \frac{23}{16} - u s = \frac{23}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{16}

$$(\frac{23}{16} - u) (\frac{23}{16} + u) = \frac{15}{16}$$

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

$$\frac{529}{256} - u^2 = \frac{15}{16}$$

Simplify the expression by subtracting \frac{529}{256} on both sides

$$-u^2 = \frac{15}{16}-\frac{529}{256} = -\frac{289}{256}$$

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

$$u^2 = \frac{289}{256} u = \pm\sqrt{\frac{289}{256}} = \pm \frac{17}{16}$$

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

$$r =\frac{23}{16} - \frac{17}{16} = 0.375 s = \frac{23}{16} + \frac{17}{16} = 2.500$$

Evaluate

$$\left(2x-5\right)\left(8x-3\right)$$