−2y^2−6y+7=0
step 2 of 2 : Use the discriminant, b2−4ac, to determine the number of solutions of the given quadratic equation. Then solve the quadratic equation using the formula y=−b±sqrt(b2−4ac)/√2a.
ok. how does the discriminant tell you about the roots? In other words, why is it called the discriminant?
To find the discriminant, we need to identify the coefficients a, b, and c in the quadratic equation −2y^2−6y+7=0.
In this equation,
a = -2
b = -6
c = 7
Now we can calculate the discriminant using the formula b^2 - 4ac:
Discriminant (D) = (-6)^2 - 4(-2)(7)
= 36 + 56
= 92
The discriminant is 92.
Next, we'll use the quadratic formula to solve the equation. The formula is:
y = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we have:
y = (-(-6) ± √( (-6)^2 - 4(-2)(7) )) / (2*(-2))
y = (6 ± √(36 + 56)) / (-4)
y = (6 ± √92) / (-4)
So the solutions to the equation −2y^2−6y+7=0 are:
y = (6 + √92) / (-4)
y = (6 - √92) / (-4)
To find the number of solutions and solve the quadratic equation −2y^2−6y+7=0 using the discriminant and the quadratic formula, follow these steps:
Step 1: Identify the values of a, b, and c in the quadratic equation. In this equation, we have a = -2, b = -6, and c = 7.
Step 2: Calculate the discriminant (b^2 - 4ac) using these values. The discriminant helps determine the number of solutions. In this case, the discriminant is: (-6)^2 - 4(-2)(7) = 36 + 56 = 92.
Step 3: Determine the number of solutions based on the value of the discriminant. If the discriminant is greater than zero (D > 0), there are two distinct real solutions. If the discriminant is equal to zero (D = 0), there is one real solution (a double root). If the discriminant is less than zero (D < 0), there are no real solutions.
In our case, the discriminant is 92, which is greater than zero. Therefore, the equation has two distinct real solutions.
Step 4: Use the quadratic formula to find the solutions of the equation. The quadratic formula is given by y = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from the equation, we have: y = (-(-6) ± √((-6)^2 - 4(-2)(7)) / (2(-2)).
Simplifying further, we get: y = (6 ± √(36 + 56)) / (-4).
Continuing the simplification, we have: y = (6 ± √92) / (-4).
Finally, the solutions are: y = (6 + √92) / (-4) and y = (6 - √92) / (-4).
You can simplify the solutions further if needed, but these are the exact solutions using the quadratic formula.