determine all the values of y that satisfy the equation y^4-13y^2+36=0
To determine all the values of y that satisfy the equation y^4 - 13y^2 + 36 = 0, we can treat it as a quadratic equation by letting z = y^2.
Substituting z into the equation, we have z^2 - 13z + 36 = 0.
Now, we can solve this quadratic equation for z. Factoring it, we have (z - 4)(z - 9) = 0.
Setting each factor equal to zero, we get z - 4 = 0 and z - 9 = 0.
Solving each equation, we find that z = 4 and z = 9.
Substituting back y^2 in place of z, we can solve for y:
For z = 4:
y^2 = 4
y = ±2
For z = 9:
y^2 = 9
y = ±3
Thus, all the values of y that satisfy the equation y^4 - 13y^2 + 36 = 0 are y = ±2 and y = ±3.
To determine all the values of y that satisfy the equation y^4-13y^2+36=0, we can use factoring.
Step 1: Substitute y^2 with a variable. Let's use z.
z = y^2
The equation becomes: z^2 - 13z + 36 = 0
Step 2: Factor the quadratic equation:
(z - 4)(z - 9) = 0
Step 3: Set each factor equal to zero and solve for z:
z - 4 = 0 or z - 9 = 0
Solving each equation:
z = 4 or z = 9
Step 4: Substitute back y^2 for z:
For z = 4: y^2 = 4
Taking the square root of both sides, we get:
y = ±2
For z = 9: y^2 = 9
Taking the square root of both sides, we get:
y = ±3
So, the values of y that satisfy the equation are y = 2, y = -2, y = 3, and y = -3.