Solve -14x^2-76x+48

$$- 14 x ^ { 2 } - 76 x + 48$$

Factor

$$2\left(-x-6\right)\left(7x-4\right)$$

Solution Steps

Factor out 2.

$$2\left(-7x^{2}-38x+24\right)$$

Consider -7x^{2}-38x+24. Factor the expression by grouping. First, the expression needs to be rewritten as -7x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-38 ab=-7\times 24=-168$$

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 -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -38.

$$a=4 b=-42$$

Rewrite -7x^{2}-38x+24 as \left(-7x^{2}+4x\right)+\left(-42x+24\right).

$$\left(-7x^{2}+4x\right)+\left(-42x+24\right)$$

Factor out -x in the first and -6 in the second group.

$$-x\left(7x-4\right)-6\left(7x-4\right)$$

Factor out common term 7x-4 by using distributive property.

$$\left(7x-4\right)\left(-x-6\right)$$

Rewrite the complete factored expression.

$$2\left(7x-4\right)\left(-x-6\right)$$

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.

$$-14x^{2}-76x+48=0$$

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.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\left(-14\right)\times 48}}{2\left(-14\right)}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\left(-14\right)\times 48}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-\left(-76\right)±\sqrt{5776+56\times 48}}{2\left(-14\right)}$$

Multiply 56 times 48.

$$x=\frac{-\left(-76\right)±\sqrt{5776+2688}}{2\left(-14\right)}$$

Add 5776 to 2688.

$$x=\frac{-\left(-76\right)±\sqrt{8464}}{2\left(-14\right)}$$

Take the square root of 8464.

$$x=\frac{-\left(-76\right)±92}{2\left(-14\right)}$$

The opposite of -76 is 76.

$$x=\frac{76±92}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{76±92}{-28}$$

Now solve the equation x=\frac{76±92}{-28} when ± is plus. Add 76 to 92.

$$x=\frac{168}{-28}$$

Divide 168 by -28.

$$x=-6$$

Now solve the equation x=\frac{76±92}{-28} when ± is minus. Subtract 92 from 76.

$$x=-\frac{16}{-28}$$

Reduce the fraction \frac{-16}{-28} to lowest terms by extracting and canceling out 4.

$$x=\frac{4}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and \frac{4}{7} for x_{2}.

$$-14x^{2}-76x+48=-14\left(x-\left(-6\right)\right)\left(x-\frac{4}{7}\right)$$

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

$$-14x^{2}-76x+48=-14\left(x+6\right)\left(x-\frac{4}{7}\right)$$

Subtract \frac{4}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-14x^{2}-76x+48=-14\left(x+6\right)\times \frac{-7x+4}{-7}$$

Cancel out 7, the greatest common factor in -14 and 7.

$$-14x^{2}-76x+48=2\left(x+6\right)\left(-7x+4\right)$$

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.

$$x ^ 2 +\frac{38}{7}x -\frac{24}{7} = 0$$

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

$$r + s = -\frac{38}{7} rs = -\frac{24}{7}$$

Two numbers r and s sum up to -\frac{38}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{38}{7} = -\frac{19}{7}. 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>

$$r = -\frac{19}{7} - u s = -\frac{19}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{24}{7}

$$(-\frac{19}{7} - u) (-\frac{19}{7} + u) = -\frac{24}{7}$$

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

$$\frac{361}{49} - u^2 = -\frac{24}{7}$$

Simplify the expression by subtracting \frac{361}{49} on both sides

$$-u^2 = -\frac{24}{7}-\frac{361}{49} = -\frac{529}{49}$$

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

$$u^2 = \frac{529}{49} u = \pm\sqrt{\frac{529}{49}} = \pm \frac{23}{7}$$

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

$$r =-\frac{19}{7} - \frac{23}{7} = -6 s = -\frac{19}{7} + \frac{23}{7} = 0.571$$

Evaluate

$$48-76x-14x^{2}$$