Solve 0=12x^2-57x+63

$$0=12 { x }^{ 2 } -57x+63$$

Solve for x

$$x=3$$

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$12x^{2}-57x+63=0$$

Divide both sides by 3.

$$4x^{2}-19x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-19 ab=4\times 21=84$$

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 84.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Calculate the sum for each pair.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

The solution is the pair that gives sum -19.

$$a=-12 b=-7$$

Rewrite 4x^{2}-19x+21 as \left(4x^{2}-12x\right)+\left(-7x+21\right).

$$\left(4x^{2}-12x\right)+\left(-7x+21\right)$$

Factor out 4x in the first and -7 in the second group.

$$4x\left(x-3\right)-7\left(x-3\right)$$

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

$$\left(x-3\right)\left(4x-7\right)$$

To find equation solutions, solve x-3=0 and 4x-7=0.

$$x=3 x=\frac{7}{4}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$12x^{2}-57x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -57 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-57\right)±\sqrt{\left(-57\right)^{2}-4\times 12\times 63}}{2\times 12}$$

Square -57.

$$x=\frac{-\left(-57\right)±\sqrt{3249-4\times 12\times 63}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-57\right)±\sqrt{3249-48\times 63}}{2\times 12}$$

Multiply -48 times 63.

$$x=\frac{-\left(-57\right)±\sqrt{3249-3024}}{2\times 12}$$

Add 3249 to -3024.

$$x=\frac{-\left(-57\right)±\sqrt{225}}{2\times 12}$$

Take the square root of 225.

$$x=\frac{-\left(-57\right)±15}{2\times 12}$$

The opposite of -57 is 57.

$$x=\frac{57±15}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{57±15}{24}$$

Now solve the equation x=\frac{57±15}{24} when ± is plus. Add 57 to 15.

$$x=\frac{72}{24}$$

Divide 72 by 24.

$$x=3$$

Now solve the equation x=\frac{57±15}{24} when ± is minus. Subtract 15 from 57.

$$x=\frac{42}{24}$$

Reduce the fraction \frac{42}{24} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=3 x=\frac{7}{4}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$12x^{2}-57x+63=0$$

Subtract 63 from both sides. Anything subtracted from zero gives its negation.

$$12x^{2}-57x=-63$$

Divide both sides by 12.

$$\frac{12x^{2}-57x}{12}=-\frac{63}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{57}{12}\right)x=-\frac{63}{12}$$

Reduce the fraction \frac{-57}{12} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{19}{4}x=-\frac{63}{12}$$

Reduce the fraction \frac{-63}{12} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{19}{4}x=-\frac{21}{4}$$

Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{4}x+\left(-\frac{19}{8}\right)^{2}=-\frac{21}{4}+\left(-\frac{19}{8}\right)^{2}$$

Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=-\frac{21}{4}+\frac{361}{64}$$

Add -\frac{21}{4} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=\frac{25}{64}$$

Factor x^{2}-\frac{19}{4}x+\frac{361}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x-\frac{19}{8}=\frac{5}{8} x-\frac{19}{8}=-\frac{5}{8}$$

Add \frac{19}{8} to both sides of the equation.

$$x=3 x=\frac{7}{4}$$