To find the roots and the vertex of the quadratic function \( y = 20x^2 + 180x - 567 \), we can use the following formulas:
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Roots: To find the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this case, \( a = 20 \), \( b = 180 \), and \( c = -567 \).
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Vertex: The vertex \( (h, k) \) of a quadratic given in standard form \( ax^2 + bx + c \) can be found using: \[ h = -\frac{b}{2a} \] and then substituting \( h \) back into the equation to find \( k \).
Let's calculate:
Step 1: Calculate the roots
Using the Quadratic Formula:
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Calculate \( b^2 - 4ac \):
\[ b^2 - 4ac = 180^2 - 4(20)(-567) \] \[ = 32400 + 45360 = 77760 \]
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Calculate the roots:
\[ x = \frac{-180 \pm \sqrt{77760}}{2(20)} \] \[ = \frac{-180 \pm 278.538}{40} \]
- First root:
\[ x_1 = \frac{-180 + 278.538}{40} = \frac{98.538}{40} \approx 2.463 \]
- Second root:
\[ x_2 = \frac{-180 - 278.538}{40} = \frac{-458.538}{40} \approx -11.463 \]
Roots:
- \( x_1 \approx 2.463 \)
- \( x_2 \approx -11.463 \)
Step 2: Calculate the vertex
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Calculate \( h \):
\[ h = -\frac{180}{2 \times 20} = -\frac{180}{40} = -4.5 \]
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Calculate \( k \) by substituting \( h \) back into the original equation:
\[ k = 20(-4.5)^2 + 180(-4.5) - 567 \] \[ = 20(20.25) - 810 - 567 \] \[ = 405 - 810 - 567 = -972 \]
Vertex:
- \( (h, k) = (-4.5, -972) \)
Summary:
- Roots: \( x_1 \approx 2.463, x_2 \approx -11.463 \)
- Vertex: \( (-4.5, -972) \)
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