Solve 77r^2+45r-18

$$77 r ^ { 2 } + 45 r - 18$$

Factor

$$\left(11r-3\right)\left(7r+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 77r^{2}+ar+br-18. To find a and b, set up a system to be solved.

$$a+b=45 ab=77\left(-18\right)=-1386$$

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 -1386.

$$-1,1386 -2,693 -3,462 -6,231 -7,198 -9,154 -11,126 -14,99 -18,77 -21,66 -22,63 -33,42$$

Calculate the sum for each pair.

$$-1+1386=1385 -2+693=691 -3+462=459 -6+231=225 -7+198=191 -9+154=145 -11+126=115 -14+99=85 -18+77=59 -21+66=45 -22+63=41 -33+42=9$$

The solution is the pair that gives sum 45.

$$a=-21 b=66$$

Rewrite 77r^{2}+45r-18 as \left(77r^{2}-21r\right)+\left(66r-18\right).

$$\left(77r^{2}-21r\right)+\left(66r-18\right)$$

Factor out 7r in the first and 6 in the second group.

$$7r\left(11r-3\right)+6\left(11r-3\right)$$

Factor out common term 11r-3 by using distributive property.

$$\left(11r-3\right)\left(7r+6\right)$$

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.

$$77r^{2}+45r-18=0$$

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.

$$r=\frac{-45±\sqrt{45^{2}-4\times 77\left(-18\right)}}{2\times 77}$$

Square 45.

$$r=\frac{-45±\sqrt{2025-4\times 77\left(-18\right)}}{2\times 77}$$

Multiply -4 times 77.

$$r=\frac{-45±\sqrt{2025-308\left(-18\right)}}{2\times 77}$$

Multiply -308 times -18.

$$r=\frac{-45±\sqrt{2025+5544}}{2\times 77}$$

Add 2025 to 5544.

$$r=\frac{-45±\sqrt{7569}}{2\times 77}$$

Take the square root of 7569.

$$r=\frac{-45±87}{2\times 77}$$

Multiply 2 times 77.

$$r=\frac{-45±87}{154}$$

Now solve the equation r=\frac{-45±87}{154} when ± is plus. Add -45 to 87.

$$r=\frac{42}{154}$$

Reduce the fraction \frac{42}{154} to lowest terms by extracting and canceling out 14.

$$r=\frac{3}{11}$$

Now solve the equation r=\frac{-45±87}{154} when ± is minus. Subtract 87 from -45.

$$r=-\frac{132}{154}$$

Reduce the fraction \frac{-132}{154} to lowest terms by extracting and canceling out 22.

$$r=-\frac{6}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{11} for x_{1} and -\frac{6}{7} for x_{2}.

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r-\left(-\frac{6}{7}\right)\right)$$

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

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r+\frac{6}{7}\right)$$

Subtract \frac{3}{11} from r by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\left(r+\frac{6}{7}\right)$$

Add \frac{6}{7} to r by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\times \frac{7r+6}{7}$$

Multiply \frac{11r-3}{11} times \frac{7r+6}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{11\times 7}$$

Multiply 11 times 7.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{77}$$

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

$$77r^{2}+45r-18=\left(11r-3\right)\left(7r+6\right)$$

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

$$x ^ 2 +\frac{45}{77}x -\frac{18}{77} = 0$$

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

$$r + s = -\frac{45}{77} rs = -\frac{18}{77}$$

Two numbers r and s sum up to -\frac{45}{77} exactly when the average of the two numbers is \frac{1}{2}*-\frac{45}{77} = -\frac{45}{154}. 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>

$$r = -\frac{45}{154} - u s = -\frac{45}{154} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{18}{77}

$$(-\frac{45}{154} - u) (-\frac{45}{154} + u) = -\frac{18}{77}$$

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

$$\frac{2025}{23716} - u^2 = -\frac{18}{77}$$

Simplify the expression by subtracting \frac{2025}{23716} on both sides

$$-u^2 = -\frac{18}{77}-\frac{2025}{23716} = -\frac{7569}{23716}$$

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

$$u^2 = \frac{7569}{23716} u = \pm\sqrt{\frac{7569}{23716}} = \pm \frac{87}{154}$$

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

$$r =-\frac{45}{154} - \frac{87}{154} = -0.857 s = -\frac{45}{154} + \frac{87}{154} = 0.273$$

Evaluate

$$\left(11r-3\right)\left(7r+6\right)$$