Solve 8x^2+14x+5

$$8 x ^ { 2 } + 14 x + 5$$

Factor

$$\left(2x+1\right)\left(4x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=14 ab=8\times 5=40$$

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 40.

$$1,40 2,20 4,10 5,8$$

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 14.

$$a=4 b=10$$

Rewrite 8x^{2}+14x+5 as \left(8x^{2}+4x\right)+\left(10x+5\right).

$$\left(8x^{2}+4x\right)+\left(10x+5\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(2x+1\right)+5\left(2x+1\right)$$

Factor out common term 2x+1 by using distributive property.

$$\left(2x+1\right)\left(4x+5\right)$$

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.

$$8x^{2}+14x+5=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 8\times 5}}{2\times 8}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 8\times 5}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-14±\sqrt{196-32\times 5}}{2\times 8}$$

Multiply -32 times 5.

$$x=\frac{-14±\sqrt{196-160}}{2\times 8}$$

Add 196 to -160.

$$x=\frac{-14±\sqrt{36}}{2\times 8}$$

Take the square root of 36.

$$x=\frac{-14±6}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-14±6}{16}$$

Now solve the equation x=\frac{-14±6}{16} when ± is plus. Add -14 to 6.

$$x=-\frac{8}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{1}{2}$$

Now solve the equation x=\frac{-14±6}{16} when ± is minus. Subtract 6 from -14.

$$x=-\frac{20}{16}$$

Reduce the fraction \frac{-20}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2} for x_{1} and -\frac{5}{4} for x_{2}.

$$8x^{2}+14x+5=8\left(x-\left(-\frac{1}{2}\right)\right)\left(x-\left(-\frac{5}{4}\right)\right)$$

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

$$8x^{2}+14x+5=8\left(x+\frac{1}{2}\right)\left(x+\frac{5}{4}\right)$$

Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+14x+5=8\times \frac{2x+1}{2}\left(x+\frac{5}{4}\right)$$

Add \frac{5}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+14x+5=8\times \frac{2x+1}{2}\times \frac{4x+5}{4}$$

Multiply \frac{2x+1}{2} times \frac{4x+5}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+14x+5=8\times \frac{\left(2x+1\right)\left(4x+5\right)}{2\times 4}$$

Multiply 2 times 4.

$$8x^{2}+14x+5=8\times \frac{\left(2x+1\right)\left(4x+5\right)}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8x^{2}+14x+5=\left(2x+1\right)\left(4x+5\right)$$

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 8

$$x ^ 2 +\frac{7}{4}x +\frac{5}{8} = 0$$

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

$$r + s = -\frac{7}{4} rs = \frac{5}{8}$$

Two numbers r and s sum up to -\frac{7}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{4} = -\frac{7}{8}. 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>

$$r = -\frac{7}{8} - u s = -\frac{7}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{8}

$$(-\frac{7}{8} - u) (-\frac{7}{8} + u) = \frac{5}{8}$$

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

$$\frac{49}{64} - u^2 = \frac{5}{8}$$

Simplify the expression by subtracting \frac{49}{64} on both sides

$$-u^2 = \frac{5}{8}-\frac{49}{64} = -\frac{9}{64}$$

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

$$u^2 = \frac{9}{64} u = \pm\sqrt{\frac{9}{64}} = \pm \frac{3}{8}$$

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

$$r =-\frac{7}{8} - \frac{3}{8} = -1.250 s = -\frac{7}{8} + \frac{3}{8} = -0.500$$

Evaluate

$$\left(2x+1\right)\left(4x+5\right)$$