Discover how IDNLearn.com can help you find the answers you need quickly and easily. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.

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.

\begin{tabular}{|c|c|}
\hline Equation & Number of Solutions \\
\hline [tex]$y=-3x^2+x+12$[/tex] & \\
\hline [tex]$y=2x^2-6x+5$[/tex] & \\
\hline [tex]$y=x^2+7x-11$[/tex] & \\
\hline [tex]$y=-x^2-8x-16$[/tex] & \\
\hline
\end{tabular}


Sagot :

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.

### Summary
\begin{tabular}{|c|c|}
\hline Equation & Number of Solutions \\
\hline[tex]$y=-3 x^2+x+12$[/tex] & 2 \\
\hline[tex]$y=2 x^2-6 x+5$[/tex] & 0 \\
\hline[tex]$y=x^2+7 x-11$[/tex] & 2 \\
\hline[tex]$y=-x^2-8 x-16$[/tex] & 1 \\
\hline
\end{tabular}

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
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.