Solve 101y^2-10y=-24

$$101 { y }^{ 2 } -10y=-24$$

Solve for y

$$y=\frac{5+\sqrt{2399}i}{101}\approx 0.04950495+0.484946412i$$

$$y=\frac{-\sqrt{2399}i+5}{101}\approx 0.04950495-0.484946412i$$

Steps Using the Quadratic Formula

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.

$$101y^{2}-10y=-24$$

Add 24 to both sides of the equation.

$$101y^{2}-10y-\left(-24\right)=-24-\left(-24\right)$$

Subtracting -24 from itself leaves 0.

$$101y^{2}-10y-\left(-24\right)=0$$

Subtract -24 from 0.

$$101y^{2}-10y+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 101 for a, -10 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 101\times 24}}{2\times 101}$$

Square -10.

$$y=\frac{-\left(-10\right)±\sqrt{100-4\times 101\times 24}}{2\times 101}$$

Multiply -4 times 101.

$$y=\frac{-\left(-10\right)±\sqrt{100-404\times 24}}{2\times 101}$$

Multiply -404 times 24.

$$y=\frac{-\left(-10\right)±\sqrt{100-9696}}{2\times 101}$$

Add 100 to -9696.

$$y=\frac{-\left(-10\right)±\sqrt{-9596}}{2\times 101}$$

Take the square root of -9596.

$$y=\frac{-\left(-10\right)±2\sqrt{2399}i}{2\times 101}$$

The opposite of -10 is 10.

$$y=\frac{10±2\sqrt{2399}i}{2\times 101}$$

Multiply 2 times 101.

$$y=\frac{10±2\sqrt{2399}i}{202}$$

Now solve the equation y=\frac{10±2\sqrt{2399}i}{202} when ± is plus. Add 10 to 2i\sqrt{2399}.

$$y=\frac{10+2\sqrt{2399}i}{202}$$

Divide 10+2i\sqrt{2399} by 202.

$$y=\frac{5+\sqrt{2399}i}{101}$$

Now solve the equation y=\frac{10±2\sqrt{2399}i}{202} when ± is minus. Subtract 2i\sqrt{2399} from 10.

$$y=\frac{-2\sqrt{2399}i+10}{202}$$

Divide 10-2i\sqrt{2399} by 202.

$$y=\frac{-\sqrt{2399}i+5}{101}$$

The equation is now solved.

$$y=\frac{5+\sqrt{2399}i}{101} y=\frac{-\sqrt{2399}i+5}{101}$$

Steps for Completing the Square

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.

$$101y^{2}-10y=-24$$

Divide both sides by 101.

$$\frac{101y^{2}-10y}{101}=-\frac{24}{101}$$

Dividing by 101 undoes the multiplication by 101.

$$y^{2}-\frac{10}{101}y=-\frac{24}{101}$$

Divide -\frac{10}{101}, the coefficient of the x term, by 2 to get -\frac{5}{101}. Then add the square of -\frac{5}{101} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{10}{101}y+\left(-\frac{5}{101}\right)^{2}=-\frac{24}{101}+\left(-\frac{5}{101}\right)^{2}$$

Square -\frac{5}{101} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{10}{101}y+\frac{25}{10201}=-\frac{24}{101}+\frac{25}{10201}$$

Add -\frac{24}{101} to \frac{25}{10201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{10}{101}y+\frac{25}{10201}=-\frac{2399}{10201}$$

Factor y^{2}-\frac{10}{101}y+\frac{25}{10201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{5}{101}\right)^{2}=-\frac{2399}{10201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{101}\right)^{2}}=\sqrt{-\frac{2399}{10201}}$$

Simplify.

$$y-\frac{5}{101}=\frac{\sqrt{2399}i}{101} y-\frac{5}{101}=-\frac{\sqrt{2399}i}{101}$$

Add \frac{5}{101} to both sides of the equation.

$$y=\frac{5+\sqrt{2399}i}{101} y=\frac{-\sqrt{2399}i+5}{101}$$