Solve 28x^2+x-2

$$28 x ^ { 2 } + x - 2$$

Factor

$$\left(4x-1\right)\left(7x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 28x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=1 ab=28\left(-2\right)=-56$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -56.

$$-1,56 -2,28 -4,14 -7,8$$

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=-7 b=8$$

Rewrite 28x^{2}+x-2 as \left(28x^{2}-7x\right)+\left(8x-2\right).

$$\left(28x^{2}-7x\right)+\left(8x-2\right)$$

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

$$7x\left(4x-1\right)+2\left(4x-1\right)$$

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

$$\left(4x-1\right)\left(7x+2\right)$$

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.

$$28x^{2}+x-2=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 28\left(-2\right)}}{2\times 28}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 28\left(-2\right)}}{2\times 28}$$

Multiply -4 times 28.

$$x=\frac{-1±\sqrt{1-112\left(-2\right)}}{2\times 28}$$

Multiply -112 times -2.

$$x=\frac{-1±\sqrt{1+224}}{2\times 28}$$

Add 1 to 224.

$$x=\frac{-1±\sqrt{225}}{2\times 28}$$

Take the square root of 225.

$$x=\frac{-1±15}{2\times 28}$$

Multiply 2 times 28.

$$x=\frac{-1±15}{56}$$

Now solve the equation x=\frac{-1±15}{56} when ± is plus. Add -1 to 15.

$$x=\frac{14}{56}$$

Reduce the fraction \frac{14}{56} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{-1±15}{56} when ± is minus. Subtract 15 from -1.

$$x=-\frac{16}{56}$$

Reduce the fraction \frac{-16}{56} to lowest terms by extracting and canceling out 8.

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

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{2}{7} for x_{2}.

$$28x^{2}+x-2=28\left(x-\frac{1}{4}\right)\left(x-\left(-\frac{2}{7}\right)\right)$$

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

$$28x^{2}+x-2=28\left(x-\frac{1}{4}\right)\left(x+\frac{2}{7}\right)$$

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

$$28x^{2}+x-2=28\times \frac{4x-1}{4}\left(x+\frac{2}{7}\right)$$

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

$$28x^{2}+x-2=28\times \frac{4x-1}{4}\times \frac{7x+2}{7}$$

Multiply \frac{4x-1}{4} times \frac{7x+2}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$28x^{2}+x-2=28\times \frac{\left(4x-1\right)\left(7x+2\right)}{4\times 7}$$

Multiply 4 times 7.

$$28x^{2}+x-2=28\times \frac{\left(4x-1\right)\left(7x+2\right)}{28}$$

Cancel out 28, the greatest common factor in 28 and 28.

$$28x^{2}+x-2=\left(4x-1\right)\left(7x+2\right)$$

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 28

$$x ^ 2 +\frac{1}{28}x -\frac{1}{14} = 0$$

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

$$r + s = -\frac{1}{28} rs = -\frac{1}{14}$$

Two numbers r and s sum up to -\frac{1}{28} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{28} = -\frac{1}{56}. 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>

$$r = -\frac{1}{56} - u s = -\frac{1}{56} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{14}

$$(-\frac{1}{56} - u) (-\frac{1}{56} + u) = -\frac{1}{14}$$

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

$$\frac{1}{3136} - u^2 = -\frac{1}{14}$$

Simplify the expression by subtracting \frac{1}{3136} on both sides

$$-u^2 = -\frac{1}{14}-\frac{1}{3136} = -\frac{225}{3136}$$

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

$$u^2 = \frac{225}{3136} u = \pm\sqrt{\frac{225}{3136}} = \pm \frac{15}{56}$$

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

$$r =-\frac{1}{56} - \frac{15}{56} = -0.286 s = -\frac{1}{56} + \frac{15}{56} = 0.250$$

Evaluate

$$28x^{2}+x-2$$