Solve -116x^2+224x+16

$$- 116 x ^ { 2 } + 224 x + 16$$

Factor

$$4\left(2-x\right)\left(29x+2\right)$$

Solution Steps

Factor out 4.

$$4\left(-29x^{2}+56x+4\right)$$

Consider -29x^{2}+56x+4. Factor the expression by grouping. First, the expression needs to be rewritten as -29x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=56 ab=-29\times 4=-116$$

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 -116.

$$-1,116 -2,58 -4,29$$

Calculate the sum for each pair.

$$-1+116=115 -2+58=56 -4+29=25$$

The solution is the pair that gives sum 56.

$$a=58 b=-2$$

Rewrite -29x^{2}+56x+4 as \left(-29x^{2}+58x\right)+\left(-2x+4\right).

$$\left(-29x^{2}+58x\right)+\left(-2x+4\right)$$

Factor out 29x in the first and 2 in the second group.

$$29x\left(-x+2\right)+2\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(29x+2\right)$$

Rewrite the complete factored expression.

$$4\left(-x+2\right)\left(29x+2\right)$$

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.

$$-116x^{2}+224x+16=0$$

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.

$$x=\frac{-224±\sqrt{224^{2}-4\left(-116\right)\times 16}}{2\left(-116\right)}$$

Square 224.

$$x=\frac{-224±\sqrt{50176-4\left(-116\right)\times 16}}{2\left(-116\right)}$$

Multiply -4 times -116.

$$x=\frac{-224±\sqrt{50176+464\times 16}}{2\left(-116\right)}$$

Multiply 464 times 16.

$$x=\frac{-224±\sqrt{50176+7424}}{2\left(-116\right)}$$

Add 50176 to 7424.

$$x=\frac{-224±\sqrt{57600}}{2\left(-116\right)}$$

Take the square root of 57600.

$$x=\frac{-224±240}{2\left(-116\right)}$$

Multiply 2 times -116.

$$x=\frac{-224±240}{-232}$$

Now solve the equation x=\frac{-224±240}{-232} when ± is plus. Add -224 to 240.

$$x=\frac{16}{-232}$$

Reduce the fraction \frac{16}{-232} to lowest terms by extracting and canceling out 8.

$$x=-\frac{2}{29}$$

Now solve the equation x=\frac{-224±240}{-232} when ± is minus. Subtract 240 from -224.

$$x=-\frac{464}{-232}$$

Divide -464 by -232.

$$x=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{29} for x_{1} and 2 for x_{2}.

$$-116x^{2}+224x+16=-116\left(x-\left(-\frac{2}{29}\right)\right)\left(x-2\right)$$

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

$$-116x^{2}+224x+16=-116\left(x+\frac{2}{29}\right)\left(x-2\right)$$

Add \frac{2}{29} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-116x^{2}+224x+16=-116\times \frac{-29x-2}{-29}\left(x-2\right)$$

Cancel out 29, the greatest common factor in -116 and 29.

$$-116x^{2}+224x+16=4\left(-29x-2\right)\left(x-2\right)$$

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.

$$x ^ 2 -\frac{56}{29}x -\frac{4}{29} = 0$$

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

$$r + s = \frac{56}{29} rs = -\frac{4}{29}$$

Two numbers r and s sum up to \frac{56}{29} exactly when the average of the two numbers is \frac{1}{2}*\frac{56}{29} = \frac{28}{29}. 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>

$$r = \frac{28}{29} - u s = \frac{28}{29} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{29}

$$(\frac{28}{29} - u) (\frac{28}{29} + u) = -\frac{4}{29}$$

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

$$\frac{784}{841} - u^2 = -\frac{4}{29}$$

Simplify the expression by subtracting \frac{784}{841} on both sides

$$-u^2 = -\frac{4}{29}-\frac{784}{841} = -\frac{900}{841}$$

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

$$u^2 = \frac{900}{841} u = \pm\sqrt{\frac{900}{841}} = \pm \frac{30}{29}$$

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

$$r =\frac{28}{29} - \frac{30}{29} = -0.069 s = \frac{28}{29} + \frac{30}{29} = 2$$

Evaluate

$$16+224x-116x^{2}$$