Solve 32x^2-4x-21

$$32 x ^ { 2 } - 4 x - 21$$

Factor

$$\left(8x-7\right)\left(4x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 32x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=-4 ab=32\left(-21\right)=-672$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -672.

$$1,-672 2,-336 3,-224 4,-168 6,-112 7,-96 8,-84 12,-56 14,-48 16,-42 21,-32 24,-28$$

Calculate the sum for each pair.

$$1-672=-671 2-336=-334 3-224=-221 4-168=-164 6-112=-106 7-96=-89 8-84=-76 12-56=-44 14-48=-34 16-42=-26 21-32=-11 24-28=-4$$

The solution is the pair that gives sum -4.

$$a=-28 b=24$$

Rewrite 32x^{2}-4x-21 as \left(32x^{2}-28x\right)+\left(24x-21\right).

$$\left(32x^{2}-28x\right)+\left(24x-21\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(8x-7\right)+3\left(8x-7\right)$$

Factor out common term 8x-7 by using distributive property.

$$\left(8x-7\right)\left(4x+3\right)$$

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.

$$32x^{2}-4x-21=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 32\left(-21\right)}}{2\times 32}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 32\left(-21\right)}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-4\right)±\sqrt{16-128\left(-21\right)}}{2\times 32}$$

Multiply -128 times -21.

$$x=\frac{-\left(-4\right)±\sqrt{16+2688}}{2\times 32}$$

Add 16 to 2688.

$$x=\frac{-\left(-4\right)±\sqrt{2704}}{2\times 32}$$

Take the square root of 2704.

$$x=\frac{-\left(-4\right)±52}{2\times 32}$$

The opposite of -4 is 4.

$$x=\frac{4±52}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{4±52}{64}$$

Now solve the equation x=\frac{4±52}{64} when ± is plus. Add 4 to 52.

$$x=\frac{56}{64}$$

Reduce the fraction \frac{56}{64} to lowest terms by extracting and canceling out 8.

$$x=\frac{7}{8}$$

Now solve the equation x=\frac{4±52}{64} when ± is minus. Subtract 52 from 4.

$$x=-\frac{48}{64}$$

Reduce the fraction \frac{-48}{64} to lowest terms by extracting and canceling out 16.

$$x=-\frac{3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{8} for x_{1} and -\frac{3}{4} for x_{2}.

$$32x^{2}-4x-21=32\left(x-\frac{7}{8}\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$32x^{2}-4x-21=32\left(x-\frac{7}{8}\right)\left(x+\frac{3}{4}\right)$$

Subtract \frac{7}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-21=32\times \frac{8x-7}{8}\left(x+\frac{3}{4}\right)$$

Add \frac{3}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-21=32\times \frac{8x-7}{8}\times \frac{4x+3}{4}$$

Multiply \frac{8x-7}{8} times \frac{4x+3}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-21=32\times \frac{\left(8x-7\right)\left(4x+3\right)}{8\times 4}$$

Multiply 8 times 4.

$$32x^{2}-4x-21=32\times \frac{\left(8x-7\right)\left(4x+3\right)}{32}$$

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

$$32x^{2}-4x-21=\left(8x-7\right)\left(4x+3\right)$$

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 32

$$x ^ 2 -\frac{1}{8}x -\frac{21}{32} = 0$$

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

$$r + s = \frac{1}{8} rs = -\frac{21}{32}$$

Two numbers r and s sum up to \frac{1}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{8} = \frac{1}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{1}{16} - u s = \frac{1}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{32}

$$(\frac{1}{16} - u) (\frac{1}{16} + u) = -\frac{21}{32}$$

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

$$\frac{1}{256} - u^2 = -\frac{21}{32}$$

Simplify the expression by subtracting \frac{1}{256} on both sides

$$-u^2 = -\frac{21}{32}-\frac{1}{256} = -\frac{169}{256}$$

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

$$u^2 = \frac{169}{256} u = \pm\sqrt{\frac{169}{256}} = \pm \frac{13}{16}$$

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

$$r =\frac{1}{16} - \frac{13}{16} = -0.750 s = \frac{1}{16} + \frac{13}{16} = 0.875$$

Evaluate

$$\left(8x-7\right)\left(4x+3\right)$$