Solve 1.6y^2=2.4y-0.9

$$1.6 { y }^{ 2 } = 2.4y-0.9$$

Solve for y

$$y=0.75$$

Steps Using the Quadratic Formula

Subtract 2.4y from both sides.

$$1.6y^{2}-2.4y=-0.9$$

Add 0.9 to both sides.

$$1.6y^{2}-2.4y+0.9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.6 for a, -2.4 for b, and 0.9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2.4\right)±\sqrt{\left(-2.4\right)^{2}-4\times 1.6\times 0.9}}{2\times 1.6}$$

Square -2.4 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-2.4\right)±\sqrt{5.76-4\times 1.6\times 0.9}}{2\times 1.6}$$

Multiply -4 times 1.6.

$$y=\frac{-\left(-2.4\right)±\sqrt{5.76-6.4\times 0.9}}{2\times 1.6}$$

Multiply -6.4 times 0.9 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-2.4\right)±\sqrt{\frac{144-144}{25}}}{2\times 1.6}$$

Add 5.76 to -5.76 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-2.4\right)±\sqrt{0}}{2\times 1.6}$$

Take the square root of 0.

$$y=-\frac{-2.4}{2\times 1.6}$$

The opposite of -2.4 is 2.4.

$$y=\frac{2.4}{2\times 1.6}$$

Multiply 2 times 1.6.

$$y=\frac{2.4}{3.2}$$

Divide 2.4 by 3.2 by multiplying 2.4 by the reciprocal of 3.2.

$$y=0.75$$

Steps for Completing the Square

Subtract 2.4y from both sides.

$$1.6y^{2}-2.4y=-0.9$$

Divide both sides of the equation by 1.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.6y^{2}-2.4y}{1.6}=-\frac{0.9}{1.6}$$

Dividing by 1.6 undoes the multiplication by 1.6.

$$y^{2}+\left(-\frac{2.4}{1.6}\right)y=-\frac{0.9}{1.6}$$

Divide -2.4 by 1.6 by multiplying -2.4 by the reciprocal of 1.6.

$$y^{2}-1.5y=-\frac{0.9}{1.6}$$

Divide -0.9 by 1.6 by multiplying -0.9 by the reciprocal of 1.6.

$$y^{2}-1.5y=-0.5625$$

Divide -1.5, the coefficient of the x term, by 2 to get -0.75. Then add the square of -0.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-1.5y+\left(-0.75\right)^{2}=-0.5625+\left(-0.75\right)^{2}$$

Square -0.75 by squaring both the numerator and the denominator of the fraction.

$$y^{2}-1.5y+0.5625=\frac{-9+9}{16}$$

Add -0.5625 to 0.5625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-1.5y+0.5625=0$$

Factor y^{2}-1.5y+0.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-0.75\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-0.75\right)^{2}}=\sqrt{0}$$

Simplify.

$$y-0.75=0 y-0.75=0$$

Add 0.75 to both sides of the equation.

$$y=0.75 y=0.75$$

The equation is now solved. Solutions are the same.

$$y=0.75$$