Solve 16x^2+58x+7

$$16 x ^ { 2 } + 58 x + 7$$

Factor

$$\left(2x+7\right)\left(8x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=58 ab=16\times 7=112$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 112.

$$1,112 2,56 4,28 7,16 8,14$$

Calculate the sum for each pair.

$$1+112=113 2+56=58 4+28=32 7+16=23 8+14=22$$

The solution is the pair that gives sum 58.

$$a=2 b=56$$

Rewrite 16x^{2}+58x+7 as \left(16x^{2}+2x\right)+\left(56x+7\right).

$$\left(16x^{2}+2x\right)+\left(56x+7\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(8x+1\right)+7\left(8x+1\right)$$

Factor out common term 8x+1 by using distributive property.

$$\left(8x+1\right)\left(2x+7\right)$$

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.

$$16x^{2}+58x+7=0$$

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.

$$x=\frac{-58±\sqrt{58^{2}-4\times 16\times 7}}{2\times 16}$$

Square 58.

$$x=\frac{-58±\sqrt{3364-4\times 16\times 7}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-58±\sqrt{3364-64\times 7}}{2\times 16}$$

Multiply -64 times 7.

$$x=\frac{-58±\sqrt{3364-448}}{2\times 16}$$

Add 3364 to -448.

$$x=\frac{-58±\sqrt{2916}}{2\times 16}$$

Take the square root of 2916.

$$x=\frac{-58±54}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-58±54}{32}$$

Now solve the equation x=\frac{-58±54}{32} when ± is plus. Add -58 to 54.

$$x=-\frac{4}{32}$$

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

$$x=-\frac{1}{8}$$

Now solve the equation x=\frac{-58±54}{32} when ± is minus. Subtract 54 from -58.

$$x=-\frac{112}{32}$$

Reduce the fraction \frac{-112}{32} to lowest terms by extracting and canceling out 16.

$$x=-\frac{7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{8} for x_{1} and -\frac{7}{2} for x_{2}.

$$16x^{2}+58x+7=16\left(x-\left(-\frac{1}{8}\right)\right)\left(x-\left(-\frac{7}{2}\right)\right)$$

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

$$16x^{2}+58x+7=16\left(x+\frac{1}{8}\right)\left(x+\frac{7}{2}\right)$$

Add \frac{1}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+58x+7=16\times \frac{8x+1}{8}\left(x+\frac{7}{2}\right)$$

Add \frac{7}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+58x+7=16\times \frac{8x+1}{8}\times \frac{2x+7}{2}$$

Multiply \frac{8x+1}{8} times \frac{2x+7}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$16x^{2}+58x+7=16\times \frac{\left(8x+1\right)\left(2x+7\right)}{8\times 2}$$

Multiply 8 times 2.

$$16x^{2}+58x+7=16\times \frac{\left(8x+1\right)\left(2x+7\right)}{16}$$

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

$$16x^{2}+58x+7=\left(8x+1\right)\left(2x+7\right)$$

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

$$x ^ 2 +\frac{29}{8}x +\frac{7}{16} = 0$$

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

$$r + s = -\frac{29}{8} rs = \frac{7}{16}$$

Two numbers r and s sum up to -\frac{29}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{29}{8} = -\frac{29}{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>

$$r = -\frac{29}{16} - u s = -\frac{29}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{16}

$$(-\frac{29}{16} - u) (-\frac{29}{16} + u) = \frac{7}{16}$$

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

$$\frac{841}{256} - u^2 = \frac{7}{16}$$

Simplify the expression by subtracting \frac{841}{256} on both sides

$$-u^2 = \frac{7}{16}-\frac{841}{256} = -\frac{729}{256}$$

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

$$u^2 = \frac{729}{256} u = \pm\sqrt{\frac{729}{256}} = \pm \frac{27}{16}$$

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

$$r =-\frac{29}{16} - \frac{27}{16} = -3.500 s = -\frac{29}{16} + \frac{27}{16} = -0.125$$

Evaluate

$$\left(2x+7\right)\left(8x+1\right)$$