Solve 32x^2+24x+4

$$32 x ^ { 2 } + 24 x + 4$$

Factor

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

Solution Steps

Factor out 4.

$$4\left(8x^{2}+6x+1\right)$$

Consider 8x^{2}+6x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=6 ab=8\times 1=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 6.

$$a=2 b=4$$

Rewrite 8x^{2}+6x+1 as \left(8x^{2}+2x\right)+\left(4x+1\right).

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

Factor out 2x in 8x^{2}+2x.

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

Factor out common term 4x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

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.

$$32x^{2}+24x+4=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 32\times 4}}{2\times 32}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 32\times 4}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-24±\sqrt{576-128\times 4}}{2\times 32}$$

Multiply -128 times 4.

$$x=\frac{-24±\sqrt{576-512}}{2\times 32}$$

Add 576 to -512.

$$x=\frac{-24±\sqrt{64}}{2\times 32}$$

Take the square root of 64.

$$x=\frac{-24±8}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{-24±8}{64}$$

Now solve the equation x=\frac{-24±8}{64} when ± is plus. Add -24 to 8.

$$x=-\frac{16}{64}$$

Reduce the fraction \frac{-16}{64} to lowest terms by extracting and canceling out 16.

$$x=-\frac{1}{4}$$

Now solve the equation x=\frac{-24±8}{64} when ± is minus. Subtract 8 from -24.

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

Reduce the fraction \frac{-32}{64} to lowest terms by extracting and canceling out 32.

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

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

$$32x^{2}+24x+4=32\left(x-\left(-\frac{1}{4}\right)\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$32x^{2}+24x+4=32\left(x+\frac{1}{4}\right)\left(x+\frac{1}{2}\right)$$

Add \frac{1}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$32x^{2}+24x+4=32\times \frac{4x+1}{4}\left(x+\frac{1}{2}\right)$$

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

$$32x^{2}+24x+4=32\times \frac{4x+1}{4}\times \frac{2x+1}{2}$$

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

$$32x^{2}+24x+4=32\times \frac{\left(4x+1\right)\left(2x+1\right)}{4\times 2}$$

Multiply 4 times 2.

$$32x^{2}+24x+4=32\times \frac{\left(4x+1\right)\left(2x+1\right)}{8}$$

Cancel out 8, the greatest common factor in 32 and 8.

$$32x^{2}+24x+4=4\left(4x+1\right)\left(2x+1\right)$$

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 32

$$x ^ 2 +\frac{3}{4}x +\frac{1}{8} = 0$$

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

$$r + s = -\frac{3}{4} rs = \frac{1}{8}$$

Two numbers r and s sum up to -\frac{3}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{4} = -\frac{3}{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>

$$r = -\frac{3}{8} - u s = -\frac{3}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{8}

$$(-\frac{3}{8} - u) (-\frac{3}{8} + u) = \frac{1}{8}$$

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

$$\frac{9}{64} - u^2 = \frac{1}{8}$$

Simplify the expression by subtracting \frac{9}{64} on both sides

$$-u^2 = \frac{1}{8}-\frac{9}{64} = -\frac{1}{64}$$

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

$$u^2 = \frac{1}{64} u = \pm\sqrt{\frac{1}{64}} = \pm \frac{1}{8}$$

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

$$r =-\frac{3}{8} - \frac{1}{8} = -0.500 s = -\frac{3}{8} + \frac{1}{8} = -0.250$$

Evaluate

$$32x^{2}+24x+4$$