Solve y^2+10y^-400=0

$${ y }^{ 2 } +10 { y }^{ } -400=0$$

Solve for y

$$y=5\sqrt{17}-5\approx 15.615528128$$

$$y=-5\sqrt{17}-5\approx -25.615528128$$

Steps Using the Quadratic Formula

Calculate y to the power of 1 and get y.

$$y^{2}+10y-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-10±\sqrt{10^{2}-4\left(-400\right)}}{2}$$

Square 10.

$$y=\frac{-10±\sqrt{100-4\left(-400\right)}}{2}$$

Multiply -4 times -400.

$$y=\frac{-10±\sqrt{100+1600}}{2}$$

Add 100 to 1600.

$$y=\frac{-10±\sqrt{1700}}{2}$$

Take the square root of 1700.

$$y=\frac{-10±10\sqrt{17}}{2}$$

Now solve the equation y=\frac{-10±10\sqrt{17}}{2} when ± is plus. Add -10 to 10\sqrt{17}.

$$y=\frac{10\sqrt{17}-10}{2}$$

Divide -10+10\sqrt{17} by 2.

$$y=5\sqrt{17}-5$$

Now solve the equation y=\frac{-10±10\sqrt{17}}{2} when ± is minus. Subtract 10\sqrt{17} from -10.

$$y=\frac{-10\sqrt{17}-10}{2}$$

Divide -10-10\sqrt{17} by 2.

$$y=-5\sqrt{17}-5$$

The equation is now solved.

$$y=5\sqrt{17}-5 y=-5\sqrt{17}-5$$

Steps for Completing the Square

Calculate y to the power of 1 and get y.

$$y^{2}+10y-400=0$$

Add 400 to both sides. Anything plus zero gives itself.

$$y^{2}+10y=400$$

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

$$y^{2}+10y+5^{2}=400+5^{2}$$

Square 5.

$$y^{2}+10y+25=400+25$$

Add 400 to 25.

$$y^{2}+10y+25=425$$

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

$$\left(y+5\right)^{2}=425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+5\right)^{2}}=\sqrt{425}$$

Simplify.

$$y+5=5\sqrt{17} y+5=-5\sqrt{17}$$

Subtract 5 from both sides of the equation.

$$y=5\sqrt{17}-5 y=-5\sqrt{17}-5$$