Solve 4x^2-81x=729

$$4 x ^ { 2 } - 81 x = 729$$

Solve for x

$$x = -\frac{27}{4} = -6\frac{3}{4} = -6.75$$

$$x=27$$

Steps Using Factoring By Grouping

Subtract 729 from both sides.

$$4x^{2}-81x-729=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-729. To find a and b, set up a system to be solved.

$$a+b=-81 ab=4\left(-729\right)=-2916$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -2916.

$$1,-2916 2,-1458 3,-972 4,-729 6,-486 9,-324 12,-243 18,-162 27,-108 36,-81 54,-54$$

Calculate the sum for each pair.

$$1-2916=-2915 2-1458=-1456 3-972=-969 4-729=-725 6-486=-480 9-324=-315 12-243=-231 18-162=-144 27-108=-81 36-81=-45 54-54=0$$

The solution is the pair that gives sum -81.

$$a=-108 b=27$$

Rewrite 4x^{2}-81x-729 as \left(4x^{2}-108x\right)+\left(27x-729\right).

$$\left(4x^{2}-108x\right)+\left(27x-729\right)$$

Factor out 4x in the first and 27 in the second group.

$$4x\left(x-27\right)+27\left(x-27\right)$$

Factor out common term x-27 by using distributive property.

$$\left(x-27\right)\left(4x+27\right)$$

To find equation solutions, solve x-27=0 and 4x+27=0.

$$x=27 x=-\frac{27}{4}$$

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.

$$4x^{2}-81x=729$$

Subtract 729 from both sides of the equation.

$$4x^{2}-81x-729=729-729$$

Subtracting 729 from itself leaves 0.

$$4x^{2}-81x-729=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -81 for b, and -729 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 4\left(-729\right)}}{2\times 4}$$

Square -81.

$$x=\frac{-\left(-81\right)±\sqrt{6561-4\times 4\left(-729\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-81\right)±\sqrt{6561-16\left(-729\right)}}{2\times 4}$$

Multiply -16 times -729.

$$x=\frac{-\left(-81\right)±\sqrt{6561+11664}}{2\times 4}$$

Add 6561 to 11664.

$$x=\frac{-\left(-81\right)±\sqrt{18225}}{2\times 4}$$

Take the square root of 18225.

$$x=\frac{-\left(-81\right)±135}{2\times 4}$$

The opposite of -81 is 81.

$$x=\frac{81±135}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{81±135}{8}$$

Now solve the equation x=\frac{81±135}{8} when ± is plus. Add 81 to 135.

$$x=\frac{216}{8}$$

Divide 216 by 8.

$$x=27$$

Now solve the equation x=\frac{81±135}{8} when ± is minus. Subtract 135 from 81.

$$x=-\frac{54}{8}$$

Reduce the fraction \frac{-54}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{27}{4}$$

The equation is now solved.

$$x=27 x=-\frac{27}{4}$$

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.

$$4x^{2}-81x=729$$

Divide both sides by 4.

$$\frac{4x^{2}-81x}{4}=\frac{729}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{81}{4}x=\frac{729}{4}$$

Divide -\frac{81}{4}, the coefficient of the x term, by 2 to get -\frac{81}{8}. Then add the square of -\frac{81}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{81}{4}x+\left(-\frac{81}{8}\right)^{2}=\frac{729}{4}+\left(-\frac{81}{8}\right)^{2}$$

Square -\frac{81}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{81}{4}x+\frac{6561}{64}=\frac{729}{4}+\frac{6561}{64}$$

Add \frac{729}{4} to \frac{6561}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{81}{4}x+\frac{6561}{64}=\frac{18225}{64}$$

Factor x^{2}-\frac{81}{4}x+\frac{6561}{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}.

$$\left(x-\frac{81}{8}\right)^{2}=\frac{18225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{81}{8}\right)^{2}}=\sqrt{\frac{18225}{64}}$$

Simplify.

$$x-\frac{81}{8}=\frac{135}{8} x-\frac{81}{8}=-\frac{135}{8}$$

Add \frac{81}{8} to both sides of the equation.

$$x=27 x=-\frac{27}{4}$$