Solve 63x^2+22x-8

$$63 x ^ { 2 } + 22 x - 8$$

Factor

$$\left(9x-2\right)\left(7x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 63x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=22 ab=63\left(-8\right)=-504$$

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 -504.

$$-1,504 -2,252 -3,168 -4,126 -6,84 -7,72 -8,63 -9,56 -12,42 -14,36 -18,28 -21,24$$

Calculate the sum for each pair.

$$-1+504=503 -2+252=250 -3+168=165 -4+126=122 -6+84=78 -7+72=65 -8+63=55 -9+56=47 -12+42=30 -14+36=22 -18+28=10 -21+24=3$$

The solution is the pair that gives sum 22.

$$a=-14 b=36$$

Rewrite 63x^{2}+22x-8 as \left(63x^{2}-14x\right)+\left(36x-8\right).

$$\left(63x^{2}-14x\right)+\left(36x-8\right)$$

Factor out 7x in the first and 4 in the second group.

$$7x\left(9x-2\right)+4\left(9x-2\right)$$

Factor out common term 9x-2 by using distributive property.

$$\left(9x-2\right)\left(7x+4\right)$$

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.

$$63x^{2}+22x-8=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 63\left(-8\right)}}{2\times 63}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 63\left(-8\right)}}{2\times 63}$$

Multiply -4 times 63.

$$x=\frac{-22±\sqrt{484-252\left(-8\right)}}{2\times 63}$$

Multiply -252 times -8.

$$x=\frac{-22±\sqrt{484+2016}}{2\times 63}$$

Add 484 to 2016.

$$x=\frac{-22±\sqrt{2500}}{2\times 63}$$

Take the square root of 2500.

$$x=\frac{-22±50}{2\times 63}$$

Multiply 2 times 63.

$$x=\frac{-22±50}{126}$$

Now solve the equation x=\frac{-22±50}{126} when ± is plus. Add -22 to 50.

$$x=\frac{28}{126}$$

Reduce the fraction \frac{28}{126} to lowest terms by extracting and canceling out 14.

$$x=\frac{2}{9}$$

Now solve the equation x=\frac{-22±50}{126} when ± is minus. Subtract 50 from -22.

$$x=-\frac{72}{126}$$

Reduce the fraction \frac{-72}{126} to lowest terms by extracting and canceling out 18.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{9} for x_{1} and -\frac{4}{7} for x_{2}.

$$63x^{2}+22x-8=63\left(x-\frac{2}{9}\right)\left(x-\left(-\frac{4}{7}\right)\right)$$

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

$$63x^{2}+22x-8=63\left(x-\frac{2}{9}\right)\left(x+\frac{4}{7}\right)$$

Subtract \frac{2}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$63x^{2}+22x-8=63\times \frac{9x-2}{9}\left(x+\frac{4}{7}\right)$$

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

$$63x^{2}+22x-8=63\times \frac{9x-2}{9}\times \frac{7x+4}{7}$$

Multiply \frac{9x-2}{9} times \frac{7x+4}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$63x^{2}+22x-8=63\times \frac{\left(9x-2\right)\left(7x+4\right)}{9\times 7}$$

Multiply 9 times 7.

$$63x^{2}+22x-8=63\times \frac{\left(9x-2\right)\left(7x+4\right)}{63}$$

Cancel out 63, the greatest common factor in 63 and 63.

$$63x^{2}+22x-8=\left(9x-2\right)\left(7x+4\right)$$

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.This is achieved by dividing both sides of the equation by 63

$$x ^ 2 +\frac{22}{63}x -\frac{8}{63} = 0$$

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

$$r + s = -\frac{22}{63} rs = -\frac{8}{63}$$

Two numbers r and s sum up to -\frac{22}{63} exactly when the average of the two numbers is \frac{1}{2}*-\frac{22}{63} = -\frac{11}{63}. 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>

$$r = -\frac{11}{63} - u s = -\frac{11}{63} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{63}

$$(-\frac{11}{63} - u) (-\frac{11}{63} + u) = -\frac{8}{63}$$

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

$$\frac{121}{3969} - u^2 = -\frac{8}{63}$$

Simplify the expression by subtracting \frac{121}{3969} on both sides

$$-u^2 = -\frac{8}{63}-\frac{121}{3969} = -\frac{625}{3969}$$

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

$$u^2 = \frac{625}{3969} u = \pm\sqrt{\frac{625}{3969}} = \pm \frac{25}{63}$$

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

$$r =-\frac{11}{63} - \frac{25}{63} = -0.571 s = -\frac{11}{63} + \frac{25}{63} = 0.222$$

Evaluate

$$\left(9x-2\right)\left(7x+4\right)$$