Solve 30s^2-19s-63

$$30 s ^ { 2 } - 19 s - 63$$

Factor

$$\left(5s-9\right)\left(6s+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 30s^{2}+as+bs-63. To find a and b, set up a system to be solved.

$$a+b=-19 ab=30\left(-63\right)=-1890$$

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 -1890.

$$1,-1890 2,-945 3,-630 5,-378 6,-315 7,-270 9,-210 10,-189 14,-135 15,-126 18,-105 21,-90 27,-70 30,-63 35,-54 42,-45$$

Calculate the sum for each pair.

$$1-1890=-1889 2-945=-943 3-630=-627 5-378=-373 6-315=-309 7-270=-263 9-210=-201 10-189=-179 14-135=-121 15-126=-111 18-105=-87 21-90=-69 27-70=-43 30-63=-33 35-54=-19 42-45=-3$$

The solution is the pair that gives sum -19.

$$a=-54 b=35$$

Rewrite 30s^{2}-19s-63 as \left(30s^{2}-54s\right)+\left(35s-63\right).

$$\left(30s^{2}-54s\right)+\left(35s-63\right)$$

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

$$6s\left(5s-9\right)+7\left(5s-9\right)$$

Factor out common term 5s-9 by using distributive property.

$$\left(5s-9\right)\left(6s+7\right)$$

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.

$$30s^{2}-19s-63=0$$

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.

$$s=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 30\left(-63\right)}}{2\times 30}$$

Square -19.

$$s=\frac{-\left(-19\right)±\sqrt{361-4\times 30\left(-63\right)}}{2\times 30}$$

Multiply -4 times 30.

$$s=\frac{-\left(-19\right)±\sqrt{361-120\left(-63\right)}}{2\times 30}$$

Multiply -120 times -63.

$$s=\frac{-\left(-19\right)±\sqrt{361+7560}}{2\times 30}$$

Add 361 to 7560.

$$s=\frac{-\left(-19\right)±\sqrt{7921}}{2\times 30}$$

Take the square root of 7921.

$$s=\frac{-\left(-19\right)±89}{2\times 30}$$

The opposite of -19 is 19.

$$s=\frac{19±89}{2\times 30}$$

Multiply 2 times 30.

$$s=\frac{19±89}{60}$$

Now solve the equation s=\frac{19±89}{60} when ± is plus. Add 19 to 89.

$$s=\frac{108}{60}$$

Reduce the fraction \frac{108}{60} to lowest terms by extracting and canceling out 12.

$$s=\frac{9}{5}$$

Now solve the equation s=\frac{19±89}{60} when ± is minus. Subtract 89 from 19.

$$s=-\frac{70}{60}$$

Reduce the fraction \frac{-70}{60} to lowest terms by extracting and canceling out 10.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{5} for x_{1} and -\frac{7}{6} for x_{2}.

$$30s^{2}-19s-63=30\left(s-\frac{9}{5}\right)\left(s-\left(-\frac{7}{6}\right)\right)$$

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

$$30s^{2}-19s-63=30\left(s-\frac{9}{5}\right)\left(s+\frac{7}{6}\right)$$

Subtract \frac{9}{5} from s by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$30s^{2}-19s-63=30\times \frac{5s-9}{5}\left(s+\frac{7}{6}\right)$$

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

$$30s^{2}-19s-63=30\times \frac{5s-9}{5}\times \frac{6s+7}{6}$$

Multiply \frac{5s-9}{5} times \frac{6s+7}{6} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$30s^{2}-19s-63=30\times \frac{\left(5s-9\right)\left(6s+7\right)}{5\times 6}$$

Multiply 5 times 6.

$$30s^{2}-19s-63=30\times \frac{\left(5s-9\right)\left(6s+7\right)}{30}$$

Cancel out 30, the greatest common factor in 30 and 30.

$$30s^{2}-19s-63=\left(5s-9\right)\left(6s+7\right)$$

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 30

$$x ^ 2 -\frac{19}{30}x -\frac{21}{10} = 0$$

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

$$r + s = \frac{19}{30} rs = -\frac{21}{10}$$

Two numbers r and s sum up to \frac{19}{30} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{30} = \frac{19}{60}. 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>

$$r = \frac{19}{60} - u s = \frac{19}{60} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{10}

$$(\frac{19}{60} - u) (\frac{19}{60} + u) = -\frac{21}{10}$$

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

$$\frac{361}{3600} - u^2 = -\frac{21}{10}$$

Simplify the expression by subtracting \frac{361}{3600} on both sides

$$-u^2 = -\frac{21}{10}-\frac{361}{3600} = -\frac{7921}{3600}$$

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

$$u^2 = \frac{7921}{3600} u = \pm\sqrt{\frac{7921}{3600}} = \pm \frac{89}{60}$$

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

$$r =\frac{19}{60} - \frac{89}{60} = -1.167 s = \frac{19}{60} + \frac{89}{60} = 1.800$$

Evaluate

$$\left(5s-9\right)\left(6s+7\right)$$