Solve 24x^2-14x+2=0quad(m

$$24 x ^ { 2 } - 14 x + 2 = 0 \quad ( m$$

Solve for x

$$x = \frac{1}{4} = 0.25$$

$$x = \frac{1}{3} = 0.3333333333333333$$

Steps Using Factoring By Grouping

Anything times zero gives zero.

$$24x^{2}-14x+2=0$$

Divide both sides by 2.

$$12x^{2}-7x+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 12x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-7 ab=12\times 1=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-4 b=-3$$

Rewrite 12x^{2}-7x+1 as \left(12x^{2}-4x\right)+\left(-3x+1\right).

$$\left(12x^{2}-4x\right)+\left(-3x+1\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(3x-1\right)-\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(4x-1\right)$$

To find equation solutions, solve 3x-1=0 and 4x-1=0.

$$x=\frac{1}{3} x=\frac{1}{4}$$

Steps Using the Quadratic Formula

Anything times zero gives zero.

$$24x^{2}-14x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -14 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 24\times 2}}{2\times 24}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 24\times 2}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-14\right)±\sqrt{196-96\times 2}}{2\times 24}$$

Multiply -96 times 2.

$$x=\frac{-\left(-14\right)±\sqrt{196-192}}{2\times 24}$$

Add 196 to -192.

$$x=\frac{-\left(-14\right)±\sqrt{4}}{2\times 24}$$

Take the square root of 4.

$$x=\frac{-\left(-14\right)±2}{2\times 24}$$

The opposite of -14 is 14.

$$x=\frac{14±2}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{14±2}{48}$$

Now solve the equation x=\frac{14±2}{48} when ± is plus. Add 14 to 2.

$$x=\frac{16}{48}$$

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

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{14±2}{48} when ± is minus. Subtract 2 from 14.

$$x=\frac{12}{48}$$

Reduce the fraction \frac{12}{48} to lowest terms by extracting and canceling out 12.

$$x=\frac{1}{4}$$

The equation is now solved.

$$x=\frac{1}{3} x=\frac{1}{4}$$

Steps for Completing the Square

Anything times zero gives zero.

$$24x^{2}-14x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$24x^{2}-14x=-2$$

Divide both sides by 24.

$$\frac{24x^{2}-14x}{24}=-\frac{2}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{14}{24}\right)x=-\frac{2}{24}$$

Reduce the fraction \frac{-14}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{12}x=-\frac{2}{24}$$

Reduce the fraction \frac{-2}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{12}x=-\frac{1}{12}$$

Divide -\frac{7}{12}, the coefficient of the x term, by 2 to get -\frac{7}{24}. Then add the square of -\frac{7}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{12}x+\left(-\frac{7}{24}\right)^{2}=-\frac{1}{12}+\left(-\frac{7}{24}\right)^{2}$$

Square -\frac{7}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{12}x+\frac{49}{576}=-\frac{1}{12}+\frac{49}{576}$$

Add -\frac{1}{12} to \frac{49}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{12}x+\frac{49}{576}=\frac{1}{576}$$

Factor x^{2}-\frac{7}{12}x+\frac{49}{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}.

$$\left(x-\frac{7}{24}\right)^{2}=\frac{1}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{24}\right)^{2}}=\sqrt{\frac{1}{576}}$$

Simplify.

$$x-\frac{7}{24}=\frac{1}{24} x-\frac{7}{24}=-\frac{1}{24}$$

Add \frac{7}{24} to both sides of the equation.

$$x=\frac{1}{3} x=\frac{1}{4}$$

Solve for m (complex solution)

$$m\in \mathrm{C}$$

$$x=\frac{1}{4}\text{ or }x=\frac{1}{3}$$

Solve for m

$$m\in \mathrm{R}$$

$$x=\frac{1}{3}\text{ or }x=\frac{1}{4}$$