Solve 3y^2=10.5y

$$3 y ^ { 2 } = 10.5 y$$

Solve for y

$$y=3.5$$

$$y=0$$

Steps Using Factoring

Subtract 10.5y from both sides.

$$3y^{2}-10.5y=0$$

Factor out y.

$$y\left(3y-10.5\right)=0$$

To find equation solutions, solve y=0 and 3y-10.5=0.

$$y=0 y=\frac{7}{2}$$

Steps Using the Quadratic Formula

Subtract 10.5y from both sides.

$$3y^{2}-10.5y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -10.5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-10.5\right)±\sqrt{\left(-10.5\right)^{2}}}{2\times 3}$$

Take the square root of \left(-10.5\right)^{2}.

$$y=\frac{-\left(-10.5\right)±\frac{21}{2}}{2\times 3}$$

The opposite of -10.5 is 10.5.

$$y=\frac{10.5±\frac{21}{2}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{10.5±\frac{21}{2}}{6}$$

Now solve the equation y=\frac{10.5±\frac{21}{2}}{6} when ± is plus. Add 10.5 to \frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{21}{6}$$

Reduce the fraction \frac{21}{6} to lowest terms by extracting and canceling out 3.

$$y=\frac{7}{2}$$

Now solve the equation y=\frac{10.5±\frac{21}{2}}{6} when ± is minus. Subtract \frac{21}{2} from 10.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{0}{6}$$

Divide 0 by 6.

$$y=0$$

The equation is now solved.

$$y=\frac{7}{2} y=0$$

Steps for Completing the Square

Subtract 10.5y from both sides.

$$3y^{2}-10.5y=0$$

Divide both sides by 3.

$$\frac{3y^{2}-10.5y}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}+\left(-\frac{10.5}{3}\right)y=\frac{0}{3}$$

Divide -10.5 by 3.

$$y^{2}-3.5y=\frac{0}{3}$$

Divide 0 by 3.

$$y^{2}-3.5y=0$$

Divide -3.5, the coefficient of the x term, by 2 to get -1.75. Then add the square of -1.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-3.5y+\left(-1.75\right)^{2}=\left(-1.75\right)^{2}$$

Square -1.75 by squaring both the numerator and the denominator of the fraction.

$$y^{2}-3.5y+3.0625=3.0625$$

Factor y^{2}-3.5y+3.0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-1.75\right)^{2}=3.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-1.75\right)^{2}}=\sqrt{3.0625}$$

Simplify.

$$y-1.75=\frac{7}{4} y-1.75=-\frac{7}{4}$$

Add 1.75 to both sides of the equation.

$$y=\frac{7}{2} y=0$$