Let's tackle each part of your question step by step.
Part 2: Completing the Square
To find the vertex and axis of symmetry for the quadratic function:
\[ y = -x^2 + 8x + 7 \]
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First, we want to rearrange the quadratic into the standard form \( y = a(x - h)^2 + k \). We will complete the square.
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Factor out the coefficient of \( x^2 \) from the quadratic term and the linear term:
\[ y = -(x^2 - 8x) + 7 \]
- Next, we complete the square inside the parentheses. To do this, take half of the coefficient of \( x \) (which is -8), square it, and add and subtract it inside the parentheses. Half of -8 is -4, and squaring it gives 16.
\[ y = -\left(x^2 - 8x + 16 - 16\right) + 7 \]
- Rewrite it:
\[ y = -\left((x - 4)^2 - 16\right) + 7 \]
- Distribute the negative sign:
\[ y = -(x - 4)^2 + 16 + 7 \] \[ y = -(x - 4)^2 + 23 \]
Now we have it in vertex form.
Vertex and Axis of Symmetry
From the expression \( y = -(x - 4)^2 + 23 \):
- The vertex \((h, k)\) is \((4, 23)\).
- The axis of symmetry is the line \( x = h \), which in this case is:
\[ x = 4 \]
Part 3: Finding the Zeros of the Quadratic Equation
To find the zeros of the quadratic equation:
\[ y = 6x^2 + 13x + 5 \]
We can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 6 \), \( b = 13 \), and \( c = 5 \).
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 13^2 - 4(6)(5) = 169 - 120 = 49 \]
- Now we apply the quadratic formula:
\[ x = \frac{-13 \pm \sqrt{49}}{2 \times 6} \]
- Since \( \sqrt{49} = 7 \), we find:
\[ x = \frac{-13 \pm 7}{12} \]
- This gives us two solutions:
- For the positive root:
\[ x = \frac{-13 + 7}{12} = \frac{-6}{12} = -\frac{1}{2} \]
- For the negative root:
\[ x = \frac{-13 - 7}{12} = \frac{-20}{12} = -\frac{5}{3} \]
Final Answers
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Vertex of \( y = -x^2 + 8x + 7 \): \((4, 23)\)
Axis of symmetry: \( x = 4 \) -
Zeros of \( y = 6x^2 + 13x + 5 \): \[ x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3} \]
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