Solve 24y^2-17y-20

$$24 y ^ { 2 } - 17 y - 20$$

Factor

$$\left(3y-4\right)\left(8y+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 24y^{2}+ay+by-20. To find a and b, set up a system to be solved.

$$a+b=-17 ab=24\left(-20\right)=-480$$

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 -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Calculate the sum for each pair.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

The solution is the pair that gives sum -17.

$$a=-32 b=15$$

Rewrite 24y^{2}-17y-20 as \left(24y^{2}-32y\right)+\left(15y-20\right).

$$\left(24y^{2}-32y\right)+\left(15y-20\right)$$

Factor out 8y in the first and 5 in the second group.

$$8y\left(3y-4\right)+5\left(3y-4\right)$$

Factor out common term 3y-4 by using distributive property.

$$\left(3y-4\right)\left(8y+5\right)$$

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.

$$24y^{2}-17y-20=0$$

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.

$$y=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 24\left(-20\right)}}{2\times 24}$$

Square -17.

$$y=\frac{-\left(-17\right)±\sqrt{289-4\times 24\left(-20\right)}}{2\times 24}$$

Multiply -4 times 24.

$$y=\frac{-\left(-17\right)±\sqrt{289-96\left(-20\right)}}{2\times 24}$$

Multiply -96 times -20.

$$y=\frac{-\left(-17\right)±\sqrt{289+1920}}{2\times 24}$$

Add 289 to 1920.

$$y=\frac{-\left(-17\right)±\sqrt{2209}}{2\times 24}$$

Take the square root of 2209.

$$y=\frac{-\left(-17\right)±47}{2\times 24}$$

The opposite of -17 is 17.

$$y=\frac{17±47}{2\times 24}$$

Multiply 2 times 24.

$$y=\frac{17±47}{48}$$

Now solve the equation y=\frac{17±47}{48} when ± is plus. Add 17 to 47.

$$y=\frac{64}{48}$$

Reduce the fraction \frac{64}{48} to lowest terms by extracting and canceling out 16.

$$y=\frac{4}{3}$$

Now solve the equation y=\frac{17±47}{48} when ± is minus. Subtract 47 from 17.

$$y=-\frac{30}{48}$$

Reduce the fraction \frac{-30}{48} to lowest terms by extracting and canceling out 6.

$$y=-\frac{5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{3} for x_{1} and -\frac{5}{8} for x_{2}.

$$24y^{2}-17y-20=24\left(y-\frac{4}{3}\right)\left(y-\left(-\frac{5}{8}\right)\right)$$

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

$$24y^{2}-17y-20=24\left(y-\frac{4}{3}\right)\left(y+\frac{5}{8}\right)$$

Subtract \frac{4}{3} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$24y^{2}-17y-20=24\times \frac{3y-4}{3}\left(y+\frac{5}{8}\right)$$

Add \frac{5}{8} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$24y^{2}-17y-20=24\times \frac{3y-4}{3}\times \frac{8y+5}{8}$$

Multiply \frac{3y-4}{3} times \frac{8y+5}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$24y^{2}-17y-20=24\times \frac{\left(3y-4\right)\left(8y+5\right)}{3\times 8}$$

Multiply 3 times 8.

$$24y^{2}-17y-20=24\times \frac{\left(3y-4\right)\left(8y+5\right)}{24}$$

Cancel out 24, the greatest common factor in 24 and 24.

$$24y^{2}-17y-20=\left(3y-4\right)\left(8y+5\right)$$

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 24

$$x ^ 2 -\frac{17}{24}x -\frac{5}{6} = 0$$

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

$$r + s = \frac{17}{24} rs = -\frac{5}{6}$$

Two numbers r and s sum up to \frac{17}{24} exactly when the average of the two numbers is \frac{1}{2}*\frac{17}{24} = \frac{17}{48}. 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>

$$r = \frac{17}{48} - u s = \frac{17}{48} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{6}

$$(\frac{17}{48} - u) (\frac{17}{48} + u) = -\frac{5}{6}$$

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

$$\frac{289}{2304} - u^2 = -\frac{5}{6}$$

Simplify the expression by subtracting \frac{289}{2304} on both sides

$$-u^2 = -\frac{5}{6}-\frac{289}{2304} = -\frac{2209}{2304}$$

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

$$u^2 = \frac{2209}{2304} u = \pm\sqrt{\frac{2209}{2304}} = \pm \frac{47}{48}$$

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

$$r =\frac{17}{48} - \frac{47}{48} = -0.625 s = \frac{17}{48} + \frac{47}{48} = 1.333$$

Evaluate

$$\left(3y-4\right)\left(8y+5\right)$$