Solve -2x^2/9+21/2=x+1/4

$$\frac{ -2 { x }^{ 2 } }{ 9 } + \frac{ 21 }{ 2 } = \frac{ x+1 }{ 4 }$$

Solve for x

$$x = \frac{3 \sqrt{1321} - 9}{16} \approx 6.252293192$$

$$x=\frac{-3\sqrt{1321}-9}{16}\approx -7.377293192$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 36, the least common multiple of 9,2,4.

$$4\left(-2\right)x^{2}+378=9\left(x+1\right)$$

Multiply 4 and -2 to get -8.

$$-8x^{2}+378=9\left(x+1\right)$$

Use the distributive property to multiply 9 by x+1.

$$-8x^{2}+378=9x+9$$

Subtract 9x from both sides.

$$-8x^{2}+378-9x=9$$

Subtract 9 from both sides.

$$-8x^{2}+378-9x-9=0$$

Subtract 9 from 378 to get 369.

$$-8x^{2}+369-9x=0$$

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.

$$-8x^{2}-9x+369=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -9 for b, and 369 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-8\right)\times 369}}{2\left(-8\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-8\right)\times 369}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-9\right)±\sqrt{81+32\times 369}}{2\left(-8\right)}$$

Multiply 32 times 369.

$$x=\frac{-\left(-9\right)±\sqrt{81+11808}}{2\left(-8\right)}$$

Add 81 to 11808.

$$x=\frac{-\left(-9\right)±\sqrt{11889}}{2\left(-8\right)}$$

Take the square root of 11889.

$$x=\frac{-\left(-9\right)±3\sqrt{1321}}{2\left(-8\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±3\sqrt{1321}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{9±3\sqrt{1321}}{-16}$$

Now solve the equation x=\frac{9±3\sqrt{1321}}{-16} when ± is plus. Add 9 to 3\sqrt{1321}.

$$x=\frac{3\sqrt{1321}+9}{-16}$$

Divide 9+3\sqrt{1321} by -16.

$$x=\frac{-3\sqrt{1321}-9}{16}$$

Now solve the equation x=\frac{9±3\sqrt{1321}}{-16} when ± is minus. Subtract 3\sqrt{1321} from 9.

$$x=\frac{9-3\sqrt{1321}}{-16}$$

Divide 9-3\sqrt{1321} by -16.

$$x=\frac{3\sqrt{1321}-9}{16}$$

The equation is now solved.

$$x=\frac{-3\sqrt{1321}-9}{16} x=\frac{3\sqrt{1321}-9}{16}$$

Steps for Completing the Square

Multiply both sides of the equation by 36, the least common multiple of 9,2,4.

$$4\left(-2\right)x^{2}+378=9\left(x+1\right)$$

Multiply 4 and -2 to get -8.

$$-8x^{2}+378=9\left(x+1\right)$$

Use the distributive property to multiply 9 by x+1.

$$-8x^{2}+378=9x+9$$

Subtract 9x from both sides.

$$-8x^{2}+378-9x=9$$

Subtract 378 from both sides.

$$-8x^{2}-9x=9-378$$

Subtract 378 from 9 to get -369.

$$-8x^{2}-9x=-369$$

Divide both sides by -8.

$$\frac{-8x^{2}-9x}{-8}=-\frac{369}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{9}{-8}\right)x=-\frac{369}{-8}$$

Divide -9 by -8.

$$x^{2}+\frac{9}{8}x=-\frac{369}{-8}$$

Divide -369 by -8.

$$x^{2}+\frac{9}{8}x=\frac{369}{8}$$

Divide \frac{9}{8}, the coefficient of the x term, by 2 to get \frac{9}{16}. Then add the square of \frac{9}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{8}x+\left(\frac{9}{16}\right)^{2}=\frac{369}{8}+\left(\frac{9}{16}\right)^{2}$$

Square \frac{9}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{8}x+\frac{81}{256}=\frac{369}{8}+\frac{81}{256}$$

Add \frac{369}{8} to \frac{81}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{8}x+\frac{81}{256}=\frac{11889}{256}$$

Factor x^{2}+\frac{9}{8}x+\frac{81}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{16}\right)^{2}=\frac{11889}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{16}\right)^{2}}=\sqrt{\frac{11889}{256}}$$

Simplify.

$$x+\frac{9}{16}=\frac{3\sqrt{1321}}{16} x+\frac{9}{16}=-\frac{3\sqrt{1321}}{16}$$

Subtract \frac{9}{16} from both sides of the equation.

$$x=\frac{3\sqrt{1321}-9}{16} x=\frac{-3\sqrt{1321}-9}{16}$$