Solve 56s^2+17s-3

$$56 s ^ { 2 } + 17 s - 3$$

Factor

$$\left(8s-1\right)\left(7s+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 56s^{2}+as+bs-3. To find a and b, set up a system to be solved.

$$a+b=17 ab=56\left(-3\right)=-168$$

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 -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Calculate the sum for each pair.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

The solution is the pair that gives sum 17.

$$a=-7 b=24$$

Rewrite 56s^{2}+17s-3 as \left(56s^{2}-7s\right)+\left(24s-3\right).

$$\left(56s^{2}-7s\right)+\left(24s-3\right)$$

Factor out 7s in the first and 3 in the second group.

$$7s\left(8s-1\right)+3\left(8s-1\right)$$

Factor out common term 8s-1 by using distributive property.

$$\left(8s-1\right)\left(7s+3\right)$$

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.

$$56s^{2}+17s-3=0$$

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.

$$s=\frac{-17±\sqrt{17^{2}-4\times 56\left(-3\right)}}{2\times 56}$$

Square 17.

$$s=\frac{-17±\sqrt{289-4\times 56\left(-3\right)}}{2\times 56}$$

Multiply -4 times 56.

$$s=\frac{-17±\sqrt{289-224\left(-3\right)}}{2\times 56}$$

Multiply -224 times -3.

$$s=\frac{-17±\sqrt{289+672}}{2\times 56}$$

Add 289 to 672.

$$s=\frac{-17±\sqrt{961}}{2\times 56}$$

Take the square root of 961.

$$s=\frac{-17±31}{2\times 56}$$

Multiply 2 times 56.

$$s=\frac{-17±31}{112}$$

Now solve the equation s=\frac{-17±31}{112} when ± is plus. Add -17 to 31.

$$s=\frac{14}{112}$$

Reduce the fraction \frac{14}{112} to lowest terms by extracting and canceling out 14.

$$s=\frac{1}{8}$$

Now solve the equation s=\frac{-17±31}{112} when ± is minus. Subtract 31 from -17.

$$s=-\frac{48}{112}$$

Reduce the fraction \frac{-48}{112} to lowest terms by extracting and canceling out 16.

$$s=-\frac{3}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{8} for x_{1} and -\frac{3}{7} for x_{2}.

$$56s^{2}+17s-3=56\left(s-\frac{1}{8}\right)\left(s-\left(-\frac{3}{7}\right)\right)$$

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

$$56s^{2}+17s-3=56\left(s-\frac{1}{8}\right)\left(s+\frac{3}{7}\right)$$

Subtract \frac{1}{8} from s by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$56s^{2}+17s-3=56\times \frac{8s-1}{8}\left(s+\frac{3}{7}\right)$$

Add \frac{3}{7} to s by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$56s^{2}+17s-3=56\times \frac{8s-1}{8}\times \frac{7s+3}{7}$$

Multiply \frac{8s-1}{8} times \frac{7s+3}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$56s^{2}+17s-3=56\times \frac{\left(8s-1\right)\left(7s+3\right)}{8\times 7}$$

Multiply 8 times 7.

$$56s^{2}+17s-3=56\times \frac{\left(8s-1\right)\left(7s+3\right)}{56}$$

Cancel out 56, the greatest common factor in 56 and 56.

$$56s^{2}+17s-3=\left(8s-1\right)\left(7s+3\right)$$

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 56

$$x ^ 2 +\frac{17}{56}x -\frac{3}{56} = 0$$

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

$$r + s = -\frac{17}{56} rs = -\frac{3}{56}$$

Two numbers r and s sum up to -\frac{17}{56} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{56} = -\frac{17}{112}. 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>

$$r = -\frac{17}{112} - u s = -\frac{17}{112} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{56}

$$(-\frac{17}{112} - u) (-\frac{17}{112} + u) = -\frac{3}{56}$$

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

$$\frac{289}{12544} - u^2 = -\frac{3}{56}$$

Simplify the expression by subtracting \frac{289}{12544} on both sides

$$-u^2 = -\frac{3}{56}-\frac{289}{12544} = -\frac{961}{12544}$$

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

$$u^2 = \frac{961}{12544} u = \pm\sqrt{\frac{961}{12544}} = \pm \frac{31}{112}$$

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

$$r =-\frac{17}{112} - \frac{31}{112} = -0.429 s = -\frac{17}{112} + \frac{31}{112} = 0.125$$

Evaluate

$$\left(8s-1\right)\left(7s+3\right)$$