Solve y-3y-4y:+9y=1

$$y - 3 y - 4 y : + 9 y = 1$$

Solve for y

$$y=\frac{3\sqrt{5}-9}{4}\approx -0.572949017$$

$$y=\frac{-3\sqrt{5}-9}{4}\approx -3.927050983$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 9.

$$9\left(y-3y\right)-4yy=9$$

Combine y and -3y to get -2y.

$$9\left(-2\right)y-4yy=9$$

Multiply 9 and -2 to get -18.

$$-18y-4yy=9$$

Multiply y and y to get y^{2}.

$$-18y-4y^{2}=9$$

Subtract 9 from both sides.

$$-18y-4y^{2}-9=0$$

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.

$$-4y^{2}-18y-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -18 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-4\right)\left(-9\right)}}{2\left(-4\right)}$$

Square -18.

$$y=\frac{-\left(-18\right)±\sqrt{324-4\left(-4\right)\left(-9\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-\left(-18\right)±\sqrt{324+16\left(-9\right)}}{2\left(-4\right)}$$

Multiply 16 times -9.

$$y=\frac{-\left(-18\right)±\sqrt{324-144}}{2\left(-4\right)}$$

Add 324 to -144.

$$y=\frac{-\left(-18\right)±\sqrt{180}}{2\left(-4\right)}$$

Take the square root of 180.

$$y=\frac{-\left(-18\right)±6\sqrt{5}}{2\left(-4\right)}$$

The opposite of -18 is 18.

$$y=\frac{18±6\sqrt{5}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{18±6\sqrt{5}}{-8}$$

Now solve the equation y=\frac{18±6\sqrt{5}}{-8} when ± is plus. Add 18 to 6\sqrt{5}.

$$y=\frac{6\sqrt{5}+18}{-8}$$

Divide 18+6\sqrt{5} by -8.

$$y=\frac{-3\sqrt{5}-9}{4}$$

Now solve the equation y=\frac{18±6\sqrt{5}}{-8} when ± is minus. Subtract 6\sqrt{5} from 18.

$$y=\frac{18-6\sqrt{5}}{-8}$$

Divide 18-6\sqrt{5} by -8.

$$y=\frac{3\sqrt{5}-9}{4}$$

The equation is now solved.

$$y=\frac{-3\sqrt{5}-9}{4} y=\frac{3\sqrt{5}-9}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 9.

$$9\left(y-3y\right)-4yy=9$$

Combine y and -3y to get -2y.

$$9\left(-2\right)y-4yy=9$$

Multiply 9 and -2 to get -18.

$$-18y-4yy=9$$

Multiply y and y to get y^{2}.

$$-18y-4y^{2}=9$$

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.

$$-4y^{2}-18y=9$$

Divide both sides by -4.

$$\frac{-4y^{2}-18y}{-4}=\frac{9}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\left(-\frac{18}{-4}\right)y=\frac{9}{-4}$$

Reduce the fraction \frac{-18}{-4} to lowest terms by extracting and canceling out 2.

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

Divide 9 by -4.

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

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{9}{2}y+\frac{81}{16}=-\frac{9}{4}+\frac{81}{16}$$

Add -\frac{9}{4} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{9}{2}y+\frac{81}{16}=\frac{45}{16}$$

Factor y^{2}+\frac{9}{2}y+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{9}{4}\right)^{2}=\frac{45}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{9}{4}\right)^{2}}=\sqrt{\frac{45}{16}}$$

Simplify.

$$y+\frac{9}{4}=\frac{3\sqrt{5}}{4} y+\frac{9}{4}=-\frac{3\sqrt{5}}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

$$y=\frac{3\sqrt{5}-9}{4} y=\frac{-3\sqrt{5}-9}{4}$$