Solve 77x^2+147x-14

$$77 x ^ { 2 } + 147 x - 14$$

Factor

$$7\left(11x-1\right)\left(x+2\right)$$

Solution Steps

Factor out 7.

$$7\left(11x^{2}+21x-2\right)$$

Consider 11x^{2}+21x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=21 ab=11\left(-2\right)=-22$$

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 -22.

$$-1,22 -2,11$$

Calculate the sum for each pair.

$$-1+22=21 -2+11=9$$

The solution is the pair that gives sum 21.

$$a=-1 b=22$$

Rewrite 11x^{2}+21x-2 as \left(11x^{2}-x\right)+\left(22x-2\right).

$$\left(11x^{2}-x\right)+\left(22x-2\right)$$

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

$$x\left(11x-1\right)+2\left(11x-1\right)$$

Factor out common term 11x-1 by using distributive property.

$$\left(11x-1\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$7\left(11x-1\right)\left(x+2\right)$$

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.

$$77x^{2}+147x-14=0$$

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.

$$x=\frac{-147±\sqrt{147^{2}-4\times 77\left(-14\right)}}{2\times 77}$$

Square 147.

$$x=\frac{-147±\sqrt{21609-4\times 77\left(-14\right)}}{2\times 77}$$

Multiply -4 times 77.

$$x=\frac{-147±\sqrt{21609-308\left(-14\right)}}{2\times 77}$$

Multiply -308 times -14.

$$x=\frac{-147±\sqrt{21609+4312}}{2\times 77}$$

Add 21609 to 4312.

$$x=\frac{-147±\sqrt{25921}}{2\times 77}$$

Take the square root of 25921.

$$x=\frac{-147±161}{2\times 77}$$

Multiply 2 times 77.

$$x=\frac{-147±161}{154}$$

Now solve the equation x=\frac{-147±161}{154} when ± is plus. Add -147 to 161.

$$x=\frac{14}{154}$$

Reduce the fraction \frac{14}{154} to lowest terms by extracting and canceling out 14.

$$x=\frac{1}{11}$$

Now solve the equation x=\frac{-147±161}{154} when ± is minus. Subtract 161 from -147.

$$x=-\frac{308}{154}$$

Divide -308 by 154.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{11} for x_{1} and -2 for x_{2}.

$$77x^{2}+147x-14=77\left(x-\frac{1}{11}\right)\left(x-\left(-2\right)\right)$$

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

$$77x^{2}+147x-14=77\left(x-\frac{1}{11}\right)\left(x+2\right)$$

Subtract \frac{1}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$77x^{2}+147x-14=77\times \frac{11x-1}{11}\left(x+2\right)$$

Cancel out 11, the greatest common factor in 77 and 11.

$$77x^{2}+147x-14=7\left(11x-1\right)\left(x+2\right)$$

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.This is achieved by dividing both sides of the equation by 77

$$x ^ 2 +\frac{21}{11}x -\frac{2}{11} = 0$$

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

$$r + s = -\frac{21}{11} rs = -\frac{2}{11}$$

Two numbers r and s sum up to -\frac{21}{11} exactly when the average of the two numbers is \frac{1}{2}*-\frac{21}{11} = -\frac{21}{22}. 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>

$$r = -\frac{21}{22} - u s = -\frac{21}{22} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{11}

$$(-\frac{21}{22} - u) (-\frac{21}{22} + u) = -\frac{2}{11}$$

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

$$\frac{441}{484} - u^2 = -\frac{2}{11}$$

Simplify the expression by subtracting \frac{441}{484} on both sides

$$-u^2 = -\frac{2}{11}-\frac{441}{484} = -\frac{529}{484}$$

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

$$u^2 = \frac{529}{484} u = \pm\sqrt{\frac{529}{484}} = \pm \frac{23}{22}$$

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

$$r =-\frac{21}{22} - \frac{23}{22} = -2 s = -\frac{21}{22} + \frac{23}{22} = 0.091$$

Evaluate

$$7\left(11x-1\right)\left(x+2\right)$$