Solve 63v^2-130v+63

$$63 v ^ { 2 } - 130 v + 63$$

Factor

$$\left(7v-9\right)\left(9v-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 63v^{2}+av+bv+63. To find a and b, set up a system to be solved.

$$a+b=-130 ab=63\times 63=3969$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 3969.

$$-1,-3969 -3,-1323 -7,-567 -9,-441 -21,-189 -27,-147 -49,-81 -63,-63$$

Calculate the sum for each pair.

$$-1-3969=-3970 -3-1323=-1326 -7-567=-574 -9-441=-450 -21-189=-210 -27-147=-174 -49-81=-130 -63-63=-126$$

The solution is the pair that gives sum -130.

$$a=-81 b=-49$$

Rewrite 63v^{2}-130v+63 as \left(63v^{2}-81v\right)+\left(-49v+63\right).

$$\left(63v^{2}-81v\right)+\left(-49v+63\right)$$

Factor out 9v in the first and -7 in the second group.

$$9v\left(7v-9\right)-7\left(7v-9\right)$$

Factor out common term 7v-9 by using distributive property.

$$\left(7v-9\right)\left(9v-7\right)$$

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.

$$63v^{2}-130v+63=0$$

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.

$$v=\frac{-\left(-130\right)±\sqrt{\left(-130\right)^{2}-4\times 63\times 63}}{2\times 63}$$

Square -130.

$$v=\frac{-\left(-130\right)±\sqrt{16900-4\times 63\times 63}}{2\times 63}$$

Multiply -4 times 63.

$$v=\frac{-\left(-130\right)±\sqrt{16900-252\times 63}}{2\times 63}$$

Multiply -252 times 63.

$$v=\frac{-\left(-130\right)±\sqrt{16900-15876}}{2\times 63}$$

Add 16900 to -15876.

$$v=\frac{-\left(-130\right)±\sqrt{1024}}{2\times 63}$$

Take the square root of 1024.

$$v=\frac{-\left(-130\right)±32}{2\times 63}$$

The opposite of -130 is 130.

$$v=\frac{130±32}{2\times 63}$$

Multiply 2 times 63.

$$v=\frac{130±32}{126}$$

Now solve the equation v=\frac{130±32}{126} when ± is plus. Add 130 to 32.

$$v=\frac{162}{126}$$

Reduce the fraction \frac{162}{126} to lowest terms by extracting and canceling out 18.

$$v=\frac{9}{7}$$

Now solve the equation v=\frac{130±32}{126} when ± is minus. Subtract 32 from 130.

$$v=\frac{98}{126}$$

Reduce the fraction \frac{98}{126} to lowest terms by extracting and canceling out 14.

$$v=\frac{7}{9}$$

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

$$63v^{2}-130v+63=63\left(v-\frac{9}{7}\right)\left(v-\frac{7}{9}\right)$$

Subtract \frac{9}{7} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$63v^{2}-130v+63=63\times \frac{7v-9}{7}\left(v-\frac{7}{9}\right)$$

Subtract \frac{7}{9} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$63v^{2}-130v+63=63\times \frac{7v-9}{7}\times \frac{9v-7}{9}$$

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

$$63v^{2}-130v+63=63\times \frac{\left(7v-9\right)\left(9v-7\right)}{7\times 9}$$

Multiply 7 times 9.

$$63v^{2}-130v+63=63\times \frac{\left(7v-9\right)\left(9v-7\right)}{63}$$

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

$$63v^{2}-130v+63=\left(7v-9\right)\left(9v-7\right)$$

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

$$x ^ 2 -\frac{130}{63}x +1 = 0$$

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

$$r + s = \frac{130}{63} rs = 1$$

Two numbers r and s sum up to \frac{130}{63} exactly when the average of the two numbers is \frac{1}{2}*\frac{130}{63} = \frac{65}{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>

$$r = \frac{65}{63} - u s = \frac{65}{63} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1

$$(\frac{65}{63} - u) (\frac{65}{63} + u) = 1$$

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

$$\frac{4225}{3969} - u^2 = 1$$

Simplify the expression by subtracting \frac{4225}{3969} on both sides

$$-u^2 = 1-\frac{4225}{3969} = -\frac{256}{3969}$$

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

$$u^2 = \frac{256}{3969} u = \pm\sqrt{\frac{256}{3969}} = \pm \frac{16}{63}$$

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

$$r =\frac{65}{63} - \frac{16}{63} = 0.778 s = \frac{65}{63} + \frac{16}{63} = 1.286$$

Evaluate

$$\left(7v-9\right)\left(9v-7\right)$$