Solve 4-3/4(m-3)=4/9(m-3)^2-4

$$4 - \frac { 3 } { 4 } ( m - 3 ) = \frac { 4 } { 9 } ( m - 3 ) ^ { 2 } - 4$$

Solve for m

$$m = \frac{3 \sqrt{2129} + 69}{32} \approx 6.481976998$$

$$m=\frac{69-3\sqrt{2129}}{32}\approx -2.169476998$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -\frac{3}{4} by m-3.

$$4-\frac{3}{4}m+\frac{9}{4}=\frac{4}{9}\left(m-3\right)^{2}-4$$

Add 4 and \frac{9}{4} to get \frac{25}{4}.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}\left(m-3\right)^{2}-4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-3\right)^{2}.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}\left(m^{2}-6m+9\right)-4$$

Use the distributive property to multiply \frac{4}{9} by m^{2}-6m+9.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}m^{2}-\frac{8}{3}m+4-4$$

Subtract 4 from 4 to get 0.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}m^{2}-\frac{8}{3}m$$

Subtract \frac{4}{9}m^{2} from both sides.

$$\frac{25}{4}-\frac{3}{4}m-\frac{4}{9}m^{2}=-\frac{8}{3}m$$

Add \frac{8}{3}m to both sides.

$$\frac{25}{4}-\frac{3}{4}m-\frac{4}{9}m^{2}+\frac{8}{3}m=0$$

Combine -\frac{3}{4}m and \frac{8}{3}m to get \frac{23}{12}m.

$$\frac{25}{4}+\frac{23}{12}m-\frac{4}{9}m^{2}=0$$

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.

$$-\frac{4}{9}m^{2}+\frac{23}{12}m+\frac{25}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{4}{9} for a, \frac{23}{12} for b, and \frac{25}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\frac{23}{12}±\sqrt{\left(\frac{23}{12}\right)^{2}-4\left(-\frac{4}{9}\right)\times \frac{25}{4}}}{2\left(-\frac{4}{9}\right)}$$

Square \frac{23}{12} by squaring both the numerator and the denominator of the fraction.

$$m=\frac{-\frac{23}{12}±\sqrt{\frac{529}{144}-4\left(-\frac{4}{9}\right)\times \frac{25}{4}}}{2\left(-\frac{4}{9}\right)}$$

Multiply -4 times -\frac{4}{9}.

$$m=\frac{-\frac{23}{12}±\sqrt{\frac{529}{144}+\frac{16}{9}\times \frac{25}{4}}}{2\left(-\frac{4}{9}\right)}$$

Multiply \frac{16}{9} times \frac{25}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$m=\frac{-\frac{23}{12}±\sqrt{\frac{529}{144}+\frac{100}{9}}}{2\left(-\frac{4}{9}\right)}$$

Add \frac{529}{144} to \frac{100}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{-\frac{23}{12}±\sqrt{\frac{2129}{144}}}{2\left(-\frac{4}{9}\right)}$$

Take the square root of \frac{2129}{144}.

$$m=\frac{-\frac{23}{12}±\frac{\sqrt{2129}}{12}}{2\left(-\frac{4}{9}\right)}$$

Multiply 2 times -\frac{4}{9}.

$$m=\frac{-\frac{23}{12}±\frac{\sqrt{2129}}{12}}{-\frac{8}{9}}$$

Now solve the equation m=\frac{-\frac{23}{12}±\frac{\sqrt{2129}}{12}}{-\frac{8}{9}} when ± is plus. Add -\frac{23}{12} to \frac{\sqrt{2129}}{12}.

$$m=\frac{\sqrt{2129}-23}{-\frac{8}{9}\times 12}$$

Divide \frac{-23+\sqrt{2129}}{12} by -\frac{8}{9} by multiplying \frac{-23+\sqrt{2129}}{12} by the reciprocal of -\frac{8}{9}.

$$m=\frac{69-3\sqrt{2129}}{32}$$

Now solve the equation m=\frac{-\frac{23}{12}±\frac{\sqrt{2129}}{12}}{-\frac{8}{9}} when ± is minus. Subtract \frac{\sqrt{2129}}{12} from -\frac{23}{12}.

$$m=\frac{-\sqrt{2129}-23}{-\frac{8}{9}\times 12}$$

Divide \frac{-23-\sqrt{2129}}{12} by -\frac{8}{9} by multiplying \frac{-23-\sqrt{2129}}{12} by the reciprocal of -\frac{8}{9}.

$$m=\frac{3\sqrt{2129}+69}{32}$$

The equation is now solved.

$$m=\frac{69-3\sqrt{2129}}{32} m=\frac{3\sqrt{2129}+69}{32}$$

Steps for Completing the Square

Use the distributive property to multiply -\frac{3}{4} by m-3.

$$4-\frac{3}{4}m+\frac{9}{4}=\frac{4}{9}\left(m-3\right)^{2}-4$$

Add 4 and \frac{9}{4} to get \frac{25}{4}.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}\left(m-3\right)^{2}-4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-3\right)^{2}.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}\left(m^{2}-6m+9\right)-4$$

Use the distributive property to multiply \frac{4}{9} by m^{2}-6m+9.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}m^{2}-\frac{8}{3}m+4-4$$

Subtract 4 from 4 to get 0.

$$\frac{25}{4}-\frac{3}{4}m=\frac{4}{9}m^{2}-\frac{8}{3}m$$

Subtract \frac{4}{9}m^{2} from both sides.

$$\frac{25}{4}-\frac{3}{4}m-\frac{4}{9}m^{2}=-\frac{8}{3}m$$

Add \frac{8}{3}m to both sides.

$$\frac{25}{4}-\frac{3}{4}m-\frac{4}{9}m^{2}+\frac{8}{3}m=0$$

Combine -\frac{3}{4}m and \frac{8}{3}m to get \frac{23}{12}m.

$$\frac{25}{4}+\frac{23}{12}m-\frac{4}{9}m^{2}=0$$

Subtract \frac{25}{4} from both sides. Anything subtracted from zero gives its negation.

$$\frac{23}{12}m-\frac{4}{9}m^{2}=-\frac{25}{4}$$

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.

$$-\frac{4}{9}m^{2}+\frac{23}{12}m=-\frac{25}{4}$$

Divide both sides of the equation by -\frac{4}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{4}{9}m^{2}+\frac{23}{12}m}{-\frac{4}{9}}=-\frac{\frac{25}{4}}{-\frac{4}{9}}$$

Dividing by -\frac{4}{9} undoes the multiplication by -\frac{4}{9}.

$$m^{2}+\frac{\frac{23}{12}}{-\frac{4}{9}}m=-\frac{\frac{25}{4}}{-\frac{4}{9}}$$

Divide \frac{23}{12} by -\frac{4}{9} by multiplying \frac{23}{12} by the reciprocal of -\frac{4}{9}.

$$m^{2}-\frac{69}{16}m=-\frac{\frac{25}{4}}{-\frac{4}{9}}$$

Divide -\frac{25}{4} by -\frac{4}{9} by multiplying -\frac{25}{4} by the reciprocal of -\frac{4}{9}.

$$m^{2}-\frac{69}{16}m=\frac{225}{16}$$

Divide -\frac{69}{16}, the coefficient of the x term, by 2 to get -\frac{69}{32}. Then add the square of -\frac{69}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{69}{16}m+\left(-\frac{69}{32}\right)^{2}=\frac{225}{16}+\left(-\frac{69}{32}\right)^{2}$$

Square -\frac{69}{32} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{69}{16}m+\frac{4761}{1024}=\frac{225}{16}+\frac{4761}{1024}$$

Add \frac{225}{16} to \frac{4761}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{69}{16}m+\frac{4761}{1024}=\frac{19161}{1024}$$

Factor m^{2}-\frac{69}{16}m+\frac{4761}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{69}{32}\right)^{2}=\frac{19161}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{69}{32}\right)^{2}}=\sqrt{\frac{19161}{1024}}$$

Simplify.

$$m-\frac{69}{32}=\frac{3\sqrt{2129}}{32} m-\frac{69}{32}=-\frac{3\sqrt{2129}}{32}$$

Add \frac{69}{32} to both sides of the equation.

$$m=\frac{3\sqrt{2129}+69}{32} m=\frac{69-3\sqrt{2129}}{32}$$