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9 answers
To solve the equation \( x^2 = 5^{20} \) by inspection, we can take the square root of both sides.
Calculating this gives:
\[ x = \pm \sqrt{5^{20}} \]
Using the property of exponents, we know that:
\[ \sqrt{5^{20}} = 5^{10} \]
Thus, the two solutions are:
\[ x = 5^{10} \quad \text{and} \quad x = -5^{10} \]
Since we are asked to enter the lesser number first, the solutions are:
\[ -5^{10}, , 5^{10} \]
So, the final answer is:
\[ -5^{10}, , 5^{10} \]
To solve the quadratic equation
\[ -3 + \frac{(x + 12)^2}{-2} = -3, \]
we first simplify the equation. Start by isolating the term with the square:
\[ \frac{(x + 12)^2}{-2} = -3 + 3. \]
This simplifies to:
\[ \frac{(x + 12)^2}{-2} = 0. \]
Next, we multiply both sides by -2 to eliminate the fraction:
\[ (x + 12)^2 = 0. \]
Now, take the square root of both sides:
\[ x + 12 = 0. \]
Solving for \(x\) gives:
\[ x = -12. \]
Since this is a quadratic equation that results in a single solution, we conclude that the solution is:
\[ -12. \]
Given that there is only one solution, it is also the smaller solution. Thus, the final answer is:
\[ -12. \]
To analyze the quadratic equation \( y = x^2 - 8x + 15 \), we will identify the key points: the vertex, x-intercepts, and y-intercept.
The x-intercepts occur where \( y = 0 \):
\[ 0 = x^2 - 8x + 15. \]
Factoring the quadratic, we look for two numbers that multiply to \( 15 \) and add to \( -8 \). These numbers are \( -3 \) and \( -5 \):
\[ 0 = (x - 3)(x - 5). \]
Setting each factor to zero gives:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3, \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5. \]
Thus, the x-intercepts are \( (3, 0) \) and \( (5, 0) \).
The y-intercept occurs where \( x = 0 \):
\[ y = 0^2 - 8(0) + 15 = 15. \]
Thus, the y-intercept is \( (0, 15) \).
The vertex of a quadratic \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a}. \] Here, \( a = 1 \) and \( b = -8 \):
\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4. \]
Now, substituting \( x = 4 \) back into the equation to find the y-coordinate of the vertex:
\[ y = (4)^2 - 8(4) + 15 = 16 - 32 + 15 = -1. \]
Thus, the vertex is \( (4, -1) \).
To find the zeros (roots) of the quadratic function \( y = 8x^2 + 14x + 3 \), we can use the quadratic formula. The steps are as follows:
The zeros of a quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, we have:
Before applying the formula, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (14)^2 - 4(8)(3) \] \[ D = 196 - 96 \] \[ D = 100 \]
Now we use the discriminant in the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting \( b \) and \( D \):
\[ x = \frac{-14 \pm \sqrt{100}}{2 \cdot 8} \] \[ x = \frac{-14 \pm 10}{16} \]
Now we calculate the two possible values for \( x \):
\[ x_1 = \frac{-14 + 10}{16} = \frac{-4}{16} = -\frac{1}{4} \]
\[ x_2 = \frac{-14 - 10}{16} = \frac{-24}{16} = -\frac{3}{2} \]
The zeros of the quadratic function \( y = 8x^2 + 14x + 3 \) are:
\[ x_1 = -\frac{1}{4}, \quad x_2 = -\frac{3}{2} \]
To determine the vertex and axis of symmetry of the quadratic function \( y = x^2 - 4x + 19 \), we can use the formulas related to the vertex of a parabola.
The axis of symmetry for a quadratic function in the form \( y = ax^2 + bx + c \) is given by the formula:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 1 \) and \( b = -4 \). Plugging in these values:
\[ x = -\frac{-4}{2(1)} = \frac{4}{2} = 2 \]
So, the axis of symmetry is:
Axis of Symmetry: \( x = 2 \)
To find the vertex, we can substitute the x-coordinate of the vertex back into the original equation. The x-coordinate of the vertex is \( x = 2 \). Now, plug this value into the equation:
\[ y = (2)^2 - 4(2) + 19 \] \[ y = 4 - 8 + 19 \] \[ y = 15 \]
Thus, the vertex of the parabola is:
Vertex: \( (2, 15) \)
