Get detailed and accurate answers to your questions on IDNLearn.com. Ask anything and receive well-informed answers from our community of experienced professionals.
The Quadratic Formula: Mastery Test
Select the correct answer from each drop-down menu.
Determine the number of real solutions for each of the given equations.
To determine the number of real solutions for each of the given quadratic equations, we analyze the discriminant (Δ), which is given by the formula: [tex]\[ \Delta = b^2 - 4ac \][/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients from the quadratic equation in the standard form [tex]\( y = ax^2 + bx + c \)[/tex].
### Equation 1: [tex]\( y = -3x^2 + x + 12 \)[/tex] For this equation: - [tex]\( a = -3 \)[/tex] - [tex]\( b = 1 \)[/tex] - [tex]\( c = 12 \)[/tex]
Calculate the discriminant: [tex]\[ \Delta = 1^2 - 4(-3)(12) = 1 + 144 = 145 \][/tex] Since [tex]\( \Delta > 0 \)[/tex], there are 2 real solutions.
### Equation 2: [tex]\( y = 2x^2 - 6x + 5 \)[/tex] For this equation: - [tex]\( a = 2 \)[/tex] - [tex]\( b = -6 \)[/tex] - [tex]\( c = 5 \)[/tex]
Calculate the discriminant: [tex]\[ \Delta = (-6)^2 - 4(2)(5) = 36 - 40 = -4 \][/tex] Since [tex]\( \Delta < 0 \)[/tex], there are 0 real solutions.
### Equation 3: [tex]\( y = x^2 + 7x - 11 \)[/tex] For this equation: - [tex]\( a = 1 \)[/tex] - [tex]\( b = 7 \)[/tex] - [tex]\( c = -11 \)[/tex]
Calculate the discriminant: [tex]\[ \Delta = 7^2 - 4(1)(-11) = 49 + 44 = 93 \][/tex] Since [tex]\( \Delta > 0 \)[/tex], there are 2 real solutions.
### Equation 4: [tex]\( y = -x^2 - 8x - 16 \)[/tex] For this equation: - [tex]\( a = -1 \)[/tex] - [tex]\( b = -8 \)[/tex] - [tex]\( c = -16 \)[/tex]
Calculate the discriminant: [tex]\[ \Delta = (-8)^2 - 4(-1)(-16) = 64 - 64 = 0 \][/tex] Since [tex]\( \Delta = 0 \)[/tex], there is 1 real solution.
Therefore, the number of real solutions for each of the given quadratic equations is: - [tex]\( y = -3 x^2 + x + 12 \)[/tex]: 2 solutions - [tex]\( y = 2 x^2 - 6 x + 5 \)[/tex]: 0 solutions - [tex]\( y = x^2 + 7 x - 11 \)[/tex]: 2 solutions - [tex]\( y = -x^2 - 8 x - 16 \)[/tex]: 1 solution
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.
