Solve 12x^2-41x+24=0

$$12 x ^ { 2 } - 41 x + 24 = 0$$

Solve for x

$$x=\frac{3}{4}=0.75$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

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 12x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-41 ab=12\times 24=288$$

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 288.

$$-1,-288 -2,-144 -3,-96 -4,-72 -6,-48 -8,-36 -9,-32 -12,-24 -16,-18$$

Calculate the sum for each pair.

$$-1-288=-289 -2-144=-146 -3-96=-99 -4-72=-76 -6-48=-54 -8-36=-44 -9-32=-41 -12-24=-36 -16-18=-34$$

The solution is the pair that gives sum -41.

$$a=-32 b=-9$$

Rewrite 12x^{2}-41x+24 as \left(12x^{2}-32x\right)+\left(-9x+24\right).

$$\left(12x^{2}-32x\right)+\left(-9x+24\right)$$

Factor out 4x in the first and -3 in the second group.

$$4x\left(3x-8\right)-3\left(3x-8\right)$$

Factor out common term 3x-8 by using distributive property.

$$\left(3x-8\right)\left(4x-3\right)$$

To find equation solutions, solve 3x-8=0 and 4x-3=0.

$$x=\frac{8}{3} x=\frac{3}{4}$$

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.

$$12x^{2}-41x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -41 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 12\times 24}}{2\times 12}$$

Square -41.

$$x=\frac{-\left(-41\right)±\sqrt{1681-4\times 12\times 24}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-41\right)±\sqrt{1681-48\times 24}}{2\times 12}$$

Multiply -48 times 24.

$$x=\frac{-\left(-41\right)±\sqrt{1681-1152}}{2\times 12}$$

Add 1681 to -1152.

$$x=\frac{-\left(-41\right)±\sqrt{529}}{2\times 12}$$

Take the square root of 529.

$$x=\frac{-\left(-41\right)±23}{2\times 12}$$

The opposite of -41 is 41.

$$x=\frac{41±23}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{41±23}{24}$$

Now solve the equation x=\frac{41±23}{24} when ± is plus. Add 41 to 23.

$$x=\frac{64}{24}$$

Reduce the fraction \frac{64}{24} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{41±23}{24} when ± is minus. Subtract 23 from 41.

$$x=\frac{18}{24}$$

Reduce the fraction \frac{18}{24} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{8}{3} x=\frac{3}{4}$$

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.

$$12x^{2}-41x+24=0$$

Subtract 24 from both sides of the equation.

$$12x^{2}-41x+24-24=-24$$

Subtracting 24 from itself leaves 0.

$$12x^{2}-41x=-24$$

Divide both sides by 12.

$$\frac{12x^{2}-41x}{12}=-\frac{24}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}-\frac{41}{12}x=-\frac{24}{12}$$

Divide -24 by 12.

$$x^{2}-\frac{41}{12}x=-2$$

Divide -\frac{41}{12}, the coefficient of the x term, by 2 to get -\frac{41}{24}. Then add the square of -\frac{41}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{41}{12}x+\left(-\frac{41}{24}\right)^{2}=-2+\left(-\frac{41}{24}\right)^{2}$$

Square -\frac{41}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{41}{12}x+\frac{1681}{576}=-2+\frac{1681}{576}$$

Add -2 to \frac{1681}{576}.

$$x^{2}-\frac{41}{12}x+\frac{1681}{576}=\frac{529}{576}$$

Factor x^{2}-\frac{41}{12}x+\frac{1681}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{41}{24}\right)^{2}=\frac{529}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{24}\right)^{2}}=\sqrt{\frac{529}{576}}$$

Simplify.

$$x-\frac{41}{24}=\frac{23}{24} x-\frac{41}{24}=-\frac{23}{24}$$

Add \frac{41}{24} to both sides of the equation.

$$x=\frac{8}{3} x=\frac{3}{4}$$

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 12

$$x ^ 2 -\frac{41}{12}x +2 = 0$$

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

$$r + s = \frac{41}{12} rs = 2$$

Two numbers r and s sum up to \frac{41}{12} exactly when the average of the two numbers is \frac{1}{2}*\frac{41}{12} = \frac{41}{24}. 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>

$$r = \frac{41}{24} - u s = \frac{41}{24} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(\frac{41}{24} - u) (\frac{41}{24} + u) = 2$$

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

$$\frac{1681}{576} - u^2 = 2$$

Simplify the expression by subtracting \frac{1681}{576} on both sides

$$-u^2 = 2-\frac{1681}{576} = -\frac{529}{576}$$

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

$$u^2 = \frac{529}{576} u = \pm\sqrt{\frac{529}{576}} = \pm \frac{23}{24}$$

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

$$r =\frac{41}{24} - \frac{23}{24} = 0.750 s = \frac{41}{24} + \frac{23}{24} = 2.667$$