Solve 4x^2=9(4x-9)

$$4 x ^ { 2 } = 9 ( 4 x - 9 )$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 9 by 4x-9.

$$4x^{2}=36x-81$$

Subtract 36x from both sides.

$$4x^{2}-36x=-81$$

Add 81 to both sides.

$$4x^{2}-36x+81=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+81. To find a and b, set up a system to be solved.

$$a+b=-36 ab=4\times 81=324$$

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 324.

$$-1,-324 -2,-162 -3,-108 -4,-81 -6,-54 -9,-36 -12,-27 -18,-18$$

Calculate the sum for each pair.

$$-1-324=-325 -2-162=-164 -3-108=-111 -4-81=-85 -6-54=-60 -9-36=-45 -12-27=-39 -18-18=-36$$

The solution is the pair that gives sum -36.

$$a=-18 b=-18$$

Rewrite 4x^{2}-36x+81 as \left(4x^{2}-18x\right)+\left(-18x+81\right).

$$\left(4x^{2}-18x\right)+\left(-18x+81\right)$$

Factor out 2x in the first and -9 in the second group.

$$2x\left(2x-9\right)-9\left(2x-9\right)$$

Factor out common term 2x-9 by using distributive property.

$$\left(2x-9\right)\left(2x-9\right)$$

Rewrite as a binomial square.

$$\left(2x-9\right)^{2}$$

To find equation solution, solve 2x-9=0.

$$x=\frac{9}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9 by 4x-9.

$$4x^{2}=36x-81$$

Subtract 36x from both sides.

$$4x^{2}-36x=-81$$

Add 81 to both sides.

$$4x^{2}-36x+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -36 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 4\times 81}}{2\times 4}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 4\times 81}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-36\right)±\sqrt{1296-16\times 81}}{2\times 4}$$

Multiply -16 times 81.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1296}}{2\times 4}$$

Add 1296 to -1296.

$$x=\frac{-\left(-36\right)±\sqrt{0}}{2\times 4}$$

Take the square root of 0.

$$x=-\frac{-36}{2\times 4}$$

The opposite of -36 is 36.

$$x=\frac{36}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{36}{8}$$

Reduce the fraction \frac{36}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 9 by 4x-9.

$$4x^{2}=36x-81$$

Subtract 36x from both sides.

$$4x^{2}-36x=-81$$

Divide both sides by 4.

$$\frac{4x^{2}-36x}{4}=-\frac{81}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{36}{4}\right)x=-\frac{81}{4}$$

Divide -36 by 4.

$$x^{2}-9x=-\frac{81}{4}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-\frac{81}{4}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=\frac{-81+81}{4}$$

Add -\frac{81}{4} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-9x+\frac{81}{4}=0$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{9}{2}=0 x-\frac{9}{2}=0$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{9}{2} x=\frac{9}{2}$$

The equation is now solved. Solutions are the same.

$$x=\frac{9}{2}$$