Solve 4/225x^2+8/25x-64/25=25/36x^2+4x+frac{224}{25}

$$\frac{ 4 }{ 225 } { x }^{ 2 } + \frac{ 8 }{ 25 } x- \frac{ 64 }{ 25 } = \frac{ 25 }{ 36 } { x }^{ 2 } +4x+ \frac{ 224 }{ 25 }$$

Solve for x (complex solution)

$$x=\frac{-552+24\sqrt{689}i}{203}\approx -2.719211823+3.103307527i$$

$$x=\frac{-24\sqrt{689}i-552}{203}\approx -2.719211823-3.103307527i$$

Steps Using the Quadratic Formula

Subtract \frac{25}{36}x^{2} from both sides.

$$\frac{4}{225}x^{2}+\frac{8}{25}x-\frac{64}{25}-\frac{25}{36}x^{2}=4x+\frac{224}{25}$$

Combine \frac{4}{225}x^{2} and -\frac{25}{36}x^{2} to get -\frac{203}{300}x^{2}.

$$-\frac{203}{300}x^{2}+\frac{8}{25}x-\frac{64}{25}=4x+\frac{224}{25}$$

Subtract 4x from both sides.

$$-\frac{203}{300}x^{2}+\frac{8}{25}x-\frac{64}{25}-4x=\frac{224}{25}$$

Combine \frac{8}{25}x and -4x to get -\frac{92}{25}x.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x-\frac{64}{25}=\frac{224}{25}$$

Subtract \frac{224}{25} from both sides.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x-\frac{64}{25}-\frac{224}{25}=0$$

Subtract \frac{224}{25} from -\frac{64}{25} to get -\frac{288}{25}.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x-\frac{288}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{203}{300} for a, -\frac{92}{25} for b, and -\frac{288}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{92}{25}\right)±\sqrt{\left(-\frac{92}{25}\right)^{2}-4\left(-\frac{203}{300}\right)\left(-\frac{288}{25}\right)}}{2\left(-\frac{203}{300}\right)}$$

Square -\frac{92}{25} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{92}{25}\right)±\sqrt{\frac{8464}{625}-4\left(-\frac{203}{300}\right)\left(-\frac{288}{25}\right)}}{2\left(-\frac{203}{300}\right)}$$

Multiply -4 times -\frac{203}{300}.

$$x=\frac{-\left(-\frac{92}{25}\right)±\sqrt{\frac{8464}{625}+\frac{203}{75}\left(-\frac{288}{25}\right)}}{2\left(-\frac{203}{300}\right)}$$

Multiply \frac{203}{75} times -\frac{288}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{92}{25}\right)±\sqrt{\frac{8464-19488}{625}}}{2\left(-\frac{203}{300}\right)}$$

Add \frac{8464}{625} to -\frac{19488}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{92}{25}\right)±\sqrt{-\frac{11024}{625}}}{2\left(-\frac{203}{300}\right)}$$

Take the square root of -\frac{11024}{625}.

$$x=\frac{-\left(-\frac{92}{25}\right)±\frac{4\sqrt{689}i}{25}}{2\left(-\frac{203}{300}\right)}$$

The opposite of -\frac{92}{25} is \frac{92}{25}.

$$x=\frac{\frac{92}{25}±\frac{4\sqrt{689}i}{25}}{2\left(-\frac{203}{300}\right)}$$

Multiply 2 times -\frac{203}{300}.

$$x=\frac{\frac{92}{25}±\frac{4\sqrt{689}i}{25}}{-\frac{203}{150}}$$

Now solve the equation x=\frac{\frac{92}{25}±\frac{4\sqrt{689}i}{25}}{-\frac{203}{150}} when ± is plus. Add \frac{92}{25} to \frac{4i\sqrt{689}}{25}.

$$x=\frac{92+4\sqrt{689}i}{-\frac{203}{150}\times 25}$$

Divide \frac{92+4i\sqrt{689}}{25} by -\frac{203}{150} by multiplying \frac{92+4i\sqrt{689}}{25} by the reciprocal of -\frac{203}{150}.

$$x=\frac{-24\sqrt{689}i-552}{203}$$

Now solve the equation x=\frac{\frac{92}{25}±\frac{4\sqrt{689}i}{25}}{-\frac{203}{150}} when ± is minus. Subtract \frac{4i\sqrt{689}}{25} from \frac{92}{25}.

$$x=\frac{-4\sqrt{689}i+92}{-\frac{203}{150}\times 25}$$

Divide \frac{92-4i\sqrt{689}}{25} by -\frac{203}{150} by multiplying \frac{92-4i\sqrt{689}}{25} by the reciprocal of -\frac{203}{150}.

$$x=\frac{-552+24\sqrt{689}i}{203}$$

The equation is now solved.

$$x=\frac{-24\sqrt{689}i-552}{203} x=\frac{-552+24\sqrt{689}i}{203}$$

Steps for Completing the Square

Subtract \frac{25}{36}x^{2} from both sides.

$$\frac{4}{225}x^{2}+\frac{8}{25}x-\frac{64}{25}-\frac{25}{36}x^{2}=4x+\frac{224}{25}$$

Combine \frac{4}{225}x^{2} and -\frac{25}{36}x^{2} to get -\frac{203}{300}x^{2}.

$$-\frac{203}{300}x^{2}+\frac{8}{25}x-\frac{64}{25}=4x+\frac{224}{25}$$

Subtract 4x from both sides.

$$-\frac{203}{300}x^{2}+\frac{8}{25}x-\frac{64}{25}-4x=\frac{224}{25}$$

Combine \frac{8}{25}x and -4x to get -\frac{92}{25}x.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x-\frac{64}{25}=\frac{224}{25}$$

Add \frac{64}{25} to both sides.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x=\frac{224}{25}+\frac{64}{25}$$

Add \frac{224}{25} and \frac{64}{25} to get \frac{288}{25}.

$$-\frac{203}{300}x^{2}-\frac{92}{25}x=\frac{288}{25}$$

Divide both sides of the equation by -\frac{203}{300}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{203}{300}x^{2}-\frac{92}{25}x}{-\frac{203}{300}}=\frac{\frac{288}{25}}{-\frac{203}{300}}$$

Dividing by -\frac{203}{300} undoes the multiplication by -\frac{203}{300}.

$$x^{2}+\left(-\frac{\frac{92}{25}}{-\frac{203}{300}}\right)x=\frac{\frac{288}{25}}{-\frac{203}{300}}$$

Divide -\frac{92}{25} by -\frac{203}{300} by multiplying -\frac{92}{25} by the reciprocal of -\frac{203}{300}.

$$x^{2}+\frac{1104}{203}x=\frac{\frac{288}{25}}{-\frac{203}{300}}$$

Divide \frac{288}{25} by -\frac{203}{300} by multiplying \frac{288}{25} by the reciprocal of -\frac{203}{300}.

$$x^{2}+\frac{1104}{203}x=-\frac{3456}{203}$$

Divide \frac{1104}{203}, the coefficient of the x term, by 2 to get \frac{552}{203}. Then add the square of \frac{552}{203} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1104}{203}x+\left(\frac{552}{203}\right)^{2}=-\frac{3456}{203}+\left(\frac{552}{203}\right)^{2}$$

Square \frac{552}{203} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1104}{203}x+\frac{304704}{41209}=-\frac{3456}{203}+\frac{304704}{41209}$$

Add -\frac{3456}{203} to \frac{304704}{41209} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1104}{203}x+\frac{304704}{41209}=-\frac{396864}{41209}$$

Factor x^{2}+\frac{1104}{203}x+\frac{304704}{41209}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{552}{203}\right)^{2}=-\frac{396864}{41209}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{552}{203}\right)^{2}}=\sqrt{-\frac{396864}{41209}}$$

Simplify.

$$x+\frac{552}{203}=\frac{24\sqrt{689}i}{203} x+\frac{552}{203}=-\frac{24\sqrt{689}i}{203}$$

Subtract \frac{552}{203} from both sides of the equation.

$$x=\frac{-552+24\sqrt{689}i}{203} x=\frac{-24\sqrt{689}i-552}{203}$$