Solve 24x^2-82x+63=0

$$24 { x }^{ 2 } -82x+63=0$$

Solve for x

$$x = \frac{7}{6} = 1\frac{1}{6} \approx 1.166666667$$

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

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.

$$24x^{2}-82x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -82 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-82\right)±\sqrt{\left(-82\right)^{2}-4\times 24\times 63}}{2\times 24}$$

Square -82.

$$x=\frac{-\left(-82\right)±\sqrt{6724-4\times 24\times 63}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-82\right)±\sqrt{6724-96\times 63}}{2\times 24}$$

Multiply -96 times 63.

$$x=\frac{-\left(-82\right)±\sqrt{6724-6048}}{2\times 24}$$

Add 6724 to -6048.

$$x=\frac{-\left(-82\right)±\sqrt{676}}{2\times 24}$$

Take the square root of 676.

$$x=\frac{-\left(-82\right)±26}{2\times 24}$$

The opposite of -82 is 82.

$$x=\frac{82±26}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{82±26}{48}$$

Now solve the equation x=\frac{82±26}{48} when ± is plus. Add 82 to 26.

$$x=\frac{108}{48}$$

Reduce the fraction \frac{108}{48} to lowest terms by extracting and canceling out 12.

$$x=\frac{9}{4}$$

Now solve the equation x=\frac{82±26}{48} when ± is minus. Subtract 26 from 82.

$$x=\frac{56}{48}$$

Reduce the fraction \frac{56}{48} to lowest terms by extracting and canceling out 8.

$$x=\frac{7}{6}$$

The equation is now solved.

$$x=\frac{9}{4} x=\frac{7}{6}$$

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.

$$24x^{2}-82x+63=0$$

Subtract 63 from both sides of the equation.

$$24x^{2}-82x+63-63=-63$$

Subtracting 63 from itself leaves 0.

$$24x^{2}-82x=-63$$

Divide both sides by 24.

$$\frac{24x^{2}-82x}{24}=-\frac{63}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{82}{24}\right)x=-\frac{63}{24}$$

Reduce the fraction \frac{-82}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{41}{12}x=-\frac{63}{24}$$

Reduce the fraction \frac{-63}{24} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{41}{12}x=-\frac{21}{8}$$

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.

$$x^{2}-\frac{41}{12}x+\left(-\frac{41}{24}\right)^{2}=-\frac{21}{8}+\left(-\frac{41}{24}\right)^{2}$$

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

$$x^{2}-\frac{41}{12}x+\frac{1681}{576}=-\frac{21}{8}+\frac{1681}{576}$$

Add -\frac{21}{8} to \frac{1681}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{41}{12}x+\frac{1681}{576}=\frac{169}{576}$$

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}.

$$\left(x-\frac{41}{24}\right)^{2}=\frac{169}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{24}\right)^{2}}=\sqrt{\frac{169}{576}}$$

Simplify.

$$x-\frac{41}{24}=\frac{13}{24} x-\frac{41}{24}=-\frac{13}{24}$$

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

$$x=\frac{9}{4} x=\frac{7}{6}$$