Answer:
1. A. Factor and use the zero-product property; x = 0, -16
B. Use the quadratic formula; y=-3-√11, -3+√11
C. Take the square root of each side; x = -6, 6
D. Complete the square; p= -2(√3 + 1). 2(√3 - 1)
2. A. Downward; coefficient of x² is negative
B. Above; k is positive
Step-by-step explanation:
1. A. x² = –16x
Factor and use the zero-product property
[tex]\begin{array}{rcl}x^{2} & = & -16x\\x^{2} + 16x & = & 0\\x(x + 16) & = &0\\x = \mathbf{0} & & x+ 16 = 0\\& & x = \mathbf{-16}\\\end{array}[/tex]
B. y² + 6y – 2 = 0
Use the quadratic formula
a = 1; b = 6; y = -2
[tex]\begin{array}{rcl}y & = & \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \\\\ & = & \dfrac{-6\pm\sqrt{6^2-4\times1\times(-2)}}{2\times1} \\\\ & = & \dfrac{-6\pm\sqrt{36+8}}{2} \\\\ & = & \dfrac{-6\pm\sqrt{44}}{2} \\\\ & = & \dfrac{-6\pm2\sqrt{11}}{2} \\\\ & = & -3\pm\sqrt{11}\\y=\mathbf{-3-\sqrt{11}} & &y= \mathbf{-3+\sqrt{11}}\\\end{array}[/tex]
C. 2a² = 72
Take the square root of each side.
[tex]\begin{array}{rcl}2a^{2} & = & 72\\a^{2} & = & 36\\a & = & \pm 6\\a= \mathbf{-6} & & a = \mathbf{6}\\\end{array}[/tex]
D. p² + 4p = 8
Complete the square.
[tex]\begin{array}{rcl}p^{2} + 4p & = & 8\\p^{2} + 4p + 4 & = & 12\\(p + 2)^{2}& = & 12\\p + 2& = & \pm \sqrt{12}\\& = & \pm 2\sqrt{3}\\p + 2 = -2\sqrt{3} & & p +2=-2\sqrt{3}\\p = -2 - 2\sqrt{3} & & p = -2 +2\sqrt{3}\\p= \mathbf{-2(\sqrt{3}+1)} & & p= \mathbf{2(\sqrt{3}-1)}\\\end{array}[/tex]
2. y = –2x² + 3x + 4
a = -2; b = 3; c = 4
A. Direction of opening
The parabola opens downward because the coefficient of x² is negative.
B. Vertex
The vertex form of a parabola is
y = a(x - h)² + k
where (h, k) are coordinates of the vertex.
The vertex will be above. on, or below the x-axis if k is positive, zero, or negative.
[tex]\begin{array}{rcl}k& = & \dfrac{4ac-b^{2}}{2a}\\\\& = & \dfrac{4\times(-2) \times 4 - 3^{2}}{2\times4}\\\\& = & \dfrac{-32 - 9}{-8}\\\\& = & \dfrac{-41}{-8}\\\\& > &\mathbf{0}\\\end{array}[/tex]
The vertex is above the x-axis because k is positive.
The graph below shows that your parabola opens downward and the vertex is above the x-axis.
