Solve 24x^2-65x+21=0

$$24 x ^ { 2 } - 65 x + 21 = 0$$

Solve for x

$$x=\frac{3}{8}=0.375$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 24x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-65 ab=24\times 21=504$$

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 504.

$$-1,-504 -2,-252 -3,-168 -4,-126 -6,-84 -7,-72 -8,-63 -9,-56 -12,-42 -14,-36 -18,-28 -21,-24$$

Calculate the sum for each pair.

$$-1-504=-505 -2-252=-254 -3-168=-171 -4-126=-130 -6-84=-90 -7-72=-79 -8-63=-71 -9-56=-65 -12-42=-54 -14-36=-50 -18-28=-46 -21-24=-45$$

The solution is the pair that gives sum -65.

$$a=-56 b=-9$$

Rewrite 24x^{2}-65x+21 as \left(24x^{2}-56x\right)+\left(-9x+21\right).

$$\left(24x^{2}-56x\right)+\left(-9x+21\right)$$

Factor out 8x in the first and -3 in the second group.

$$8x\left(3x-7\right)-3\left(3x-7\right)$$

Factor out common term 3x-7 by using distributive property.

$$\left(3x-7\right)\left(8x-3\right)$$

To find equation solutions, solve 3x-7=0 and 8x-3=0.

$$x=\frac{7}{3} x=\frac{3}{8}$$

Steps Using the Quadratic Formula

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.

$$24x^{2}-65x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -65 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-65\right)±\sqrt{\left(-65\right)^{2}-4\times 24\times 21}}{2\times 24}$$

Square -65.

$$x=\frac{-\left(-65\right)±\sqrt{4225-4\times 24\times 21}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-65\right)±\sqrt{4225-96\times 21}}{2\times 24}$$

Multiply -96 times 21.

$$x=\frac{-\left(-65\right)±\sqrt{4225-2016}}{2\times 24}$$

Add 4225 to -2016.

$$x=\frac{-\left(-65\right)±\sqrt{2209}}{2\times 24}$$

Take the square root of 2209.

$$x=\frac{-\left(-65\right)±47}{2\times 24}$$

The opposite of -65 is 65.

$$x=\frac{65±47}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{65±47}{48}$$

Now solve the equation x=\frac{65±47}{48} when ± is plus. Add 65 to 47.

$$x=\frac{112}{48}$$

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

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{65±47}{48} when ± is minus. Subtract 47 from 65.

$$x=\frac{18}{48}$$

Reduce the fraction \frac{18}{48} to lowest terms by extracting and canceling out 6.

$$x=\frac{3}{8}$$

The equation is now solved.

$$x=\frac{7}{3} x=\frac{3}{8}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$24x^{2}-65x+21=0$$

Subtract 21 from both sides of the equation.

$$24x^{2}-65x+21-21=-21$$

Subtracting 21 from itself leaves 0.

$$24x^{2}-65x=-21$$

Divide both sides by 24.

$$\frac{24x^{2}-65x}{24}=-\frac{21}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}-\frac{65}{24}x=-\frac{21}{24}$$

Reduce the fraction \frac{-21}{24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{65}{24}x=-\frac{7}{8}$$

Divide -\frac{65}{24}, the coefficient of the x term, by 2 to get -\frac{65}{48}. Then add the square of -\frac{65}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{65}{24}x+\left(-\frac{65}{48}\right)^{2}=-\frac{7}{8}+\left(-\frac{65}{48}\right)^{2}$$

Square -\frac{65}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{65}{24}x+\frac{4225}{2304}=-\frac{7}{8}+\frac{4225}{2304}$$

Add -\frac{7}{8} to \frac{4225}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{65}{24}x+\frac{4225}{2304}=\frac{2209}{2304}$$

Factor x^{2}-\frac{65}{24}x+\frac{4225}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{65}{48}\right)^{2}=\frac{2209}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{65}{48}\right)^{2}}=\sqrt{\frac{2209}{2304}}$$

Simplify.

$$x-\frac{65}{48}=\frac{47}{48} x-\frac{65}{48}=-\frac{47}{48}$$

Add \frac{65}{48} to both sides of the equation.

$$x=\frac{7}{3} x=\frac{3}{8}$$

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

$$x ^ 2 -\frac{65}{24}x +\frac{7}{8} = 0$$

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

$$r + s = \frac{65}{24} rs = \frac{7}{8}$$

Two numbers r and s sum up to \frac{65}{24} exactly when the average of the two numbers is \frac{1}{2}*\frac{65}{24} = \frac{65}{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>

$$r = \frac{65}{48} - u s = \frac{65}{48} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{8}

$$(\frac{65}{48} - u) (\frac{65}{48} + u) = \frac{7}{8}$$

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

$$\frac{4225}{2304} - u^2 = \frac{7}{8}$$

Simplify the expression by subtracting \frac{4225}{2304} on both sides

$$-u^2 = \frac{7}{8}-\frac{4225}{2304} = -\frac{2209}{2304}$$

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.

$$r =\frac{65}{48} - \frac{47}{48} = 0.375 s = \frac{65}{48} + \frac{47}{48} = 2.333$$